From bb36abed1a1e05d8b79a227cd77c452e3e1433dc Mon Sep 17 00:00:00 2001 From: GitHub Action Date: Sun, 8 Sep 2024 10:29:25 +0000 Subject: [PATCH] Update documentation --- .../plot_gfm_rfpsc_13kva.ipynb | 4 +- .../grid_examples_jupyter.zip | Bin 22820 -> 22864 bytes .../plot_vhz_ctrl_im_2kw.ipynb | 2 +- .../drive_examples_jupyter.zip | Bin 113456 -> 113451 bytes .../plot_flux_vector_pmsyrm_5kw.zip | Bin 17263 -> 17263 bytes .../plot_vector_ctrl_im_2kw.zip | Bin 9634 -> 9634 bytes .../plot_gfm_obs_13kva.zip | Bin 7246 -> 7248 bytes .../plot_gfm_rfpsc_13kva.py | 9 +- .../plot_vector_ctrl_im_2kw_tq_mode.zip | Bin 6829 -> 6829 bytes .../plot_gfm_obs_13kva.py | 6 +- .../plot_gfl_10kva.zip | Bin 6913 -> 6889 bytes .../plot_obs_vhz_ctrl_pmsm_2kw_two_mass.zip | Bin 11021 -> 11021 bytes .../plot_gfl_dc_bus_10kva.zip | Bin 6773 -> 6773 bytes .../plot_vector_ctrl_pmsm_2kw_diode.zip | Bin 5996 -> 5996 bytes .../plot_vhz_ctrl_6step_im_2kw.zip | Bin 7385 -> 7377 bytes .../plot_vector_ctrl_pmsyrm_thor.zip | Bin 7278 -> 7278 bytes .../plot_obs_vhz_ctrl_im_2kw.zip | Bin 7037 -> 7037 bytes .../plot_flux_vector_pmsm_2kw.zip | Bin 5614 -> 5614 bytes .../plot_gfm_obs_13kva.ipynb | 4 +- .../plot_vector_ctrl_pmsm_2kw.zip | Bin 5654 -> 5654 bytes .../plot_vhz_ctrl_6step_im_2kw.py | 8 +- .../plot_vhz_ctrl_im_2kw.py | 2 +- .../plot_obs_vhz_ctrl_pmsm_2kw.zip | Bin 6263 -> 6263 bytes .../grid_examples_python.zip | Bin 10485 -> 10526 bytes .../plot_gfl_10kva.py | 4 +- .../plot_gfl_lcl_10kva.zip | Bin 6010 -> 6010 bytes .../plot_vhz_ctrl_im_2kw.zip | Bin 8262 -> 8260 bytes .../plot_signal_inj_pmsm_2kw.zip | Bin 7385 -> 7385 bytes .../plot_gfl_10kva.ipynb | 2 +- .../plot_gfm_rfpsc_13kva.zip | Bin 6145 -> 6252 bytes .../plot_signal_inj_syrm_7kw.zip | Bin 7582 -> 7582 bytes .../plot_vhz_ctrl_im_2kw_lc.zip | Bin 9379 -> 9379 bytes .../plot_obs_vhz_ctrl_pmsyrm_thor.zip | Bin 12129 -> 12129 bytes .../plot_flux_vector_syrm_7kw.zip | Bin 13340 -> 13340 bytes .../plot_obs_vhz_ctrl_syrm_7kw.zip | Bin 12747 -> 12747 bytes .../drive_examples_python.zip | Bin 60279 -> 60274 bytes .../plot_flux_vector_im_2kw.zip | Bin 6711 -> 6711 bytes .../plot_vector_ctrl_syrm_7kw.zip | Bin 5966 -> 5966 bytes .../plot_vhz_ctrl_6step_im_2kw.ipynb | 2 +- .../sphx_glr_plot_gfm_obs_13kva_001.png | Bin 461295 -> 461296 bytes .../sphx_glr_plot_gfm_obs_13kva_002.png | Bin 72664 -> 70793 bytes .../sphx_glr_plot_gfm_obs_13kva_thumb.png | Bin 152454 -> 152456 bytes .../sphx_glr_plot_gfm_rfpsc_13kva_001.png | Bin 459569 -> 463312 bytes .../sphx_glr_plot_gfm_rfpsc_13kva_002.png | Bin 73423 -> 69480 bytes .../sphx_glr_plot_gfm_rfpsc_13kva_thumb.png | Bin 154176 -> 154412 bytes docs/_modules/index.html | 3 +- .../motulator/common/control/_control.html | 3 +- .../motulator/common/model/_converter.html | 3 +- .../motulator/common/model/_simulation.html | 3 +- .../motulator/common/utils/_utils.html | 3 +- .../motulator/drive/control/_common.html | 3 +- .../motulator/drive/control/im/_common.html | 3 +- .../drive/control/im/_current_vector.html | 3 +- .../drive/control/im/_flux_vector.html | 3 +- .../motulator/drive/control/im/_obs_vhz.html | 3 +- .../motulator/drive/control/im/_vhz.html | 9 +- .../motulator/drive/control/sm/_common.html | 3 +- .../drive/control/sm/_current_vector.html | 3 +- .../drive/control/sm/_flux_vector.html | 33 +- .../motulator/drive/control/sm/_obs_vhz.html | 3 +- .../drive/control/sm/_signal_inj.html | 3 +- .../motulator/drive/control/sm/_torque.html | 3 +- .../motulator/drive/model/_drive.html | 3 +- .../motulator/drive/model/_lc_filter.html | 3 +- .../motulator/drive/model/_machine.html | 3 +- .../motulator/drive/model/_mechanics.html | 3 +- .../motulator/drive/utils/_flux_maps.html | 3 +- .../motulator/drive/utils/_helpers.html | 3 +- .../motulator/drive/utils/_plots.html | 3 +- .../motulator/grid/control/_common.html | 49 +- .../grid/control/_grid_following.html | 6 +- .../motulator/grid/control/_observer_gfm.html | 84 ++- .../grid/control/_power_synchronization.html | 24 +- .../motulator/grid/model/_ac_filter.html | 3 +- .../grid/model/_converter_system.html | 3 +- .../motulator/grid/model/_voltage_source.html | 3 +- .../_modules/motulator/grid/utils/_plots.html | 16 +- .../_modules/motulator/grid/utils/_utils.html | 3 +- .../autoapi/motulator/grid/control/index.rst | 22 +- docs/_sources/control/control_system.rst | 30 +- docs/_sources/control/drive/current_ctrl.rst | 46 +- docs/_sources/control/drive/observers.rst | 36 +- docs/_sources/control/grid/current_ctrl.rst | 25 +- .../_sources/control/grid/dc_voltage_ctrl.rst | 16 +- docs/_sources/control/grid/grid-forming.rst | 92 +++ docs/_sources/control/grid/index.rst | 5 +- docs/_sources/control/grid/pll.rst | 100 +++ .../drive_examples/flux_vector/index.rst | 2 +- .../flux_vector/plot_flux_vector_im_2kw.rst | 2 +- .../flux_vector/plot_flux_vector_pmsm_2kw.rst | 2 +- .../plot_flux_vector_pmsyrm_5kw.rst | 2 +- .../flux_vector/plot_flux_vector_syrm_7kw.rst | 2 +- .../flux_vector/sg_execution_times.rst | 10 +- docs/_sources/drive_examples/index.rst | 6 +- .../obs_vhz/plot_obs_vhz_ctrl_im_2kw.rst | 2 +- .../obs_vhz/plot_obs_vhz_ctrl_pmsm_2kw.rst | 2 +- .../plot_obs_vhz_ctrl_pmsm_2kw_two_mass.rst | 2 +- .../obs_vhz/plot_obs_vhz_ctrl_pmsyrm_thor.rst | 2 +- .../obs_vhz/plot_obs_vhz_ctrl_syrm_7kw.rst | 2 +- .../obs_vhz/sg_execution_times.rst | 12 +- .../signal_inj/plot_signal_inj_pmsm_2kw.rst | 2 +- .../signal_inj/plot_signal_inj_syrm_7kw.rst | 2 +- .../signal_inj/sg_execution_times.rst | 6 +- .../vector/plot_vector_ctrl_im_2kw.rst | 2 +- .../plot_vector_ctrl_im_2kw_tq_mode.rst | 2 +- .../vector/plot_vector_ctrl_pmsm_2kw.rst | 4 +- .../plot_vector_ctrl_pmsm_2kw_diode.rst | 2 +- .../vector/plot_vector_ctrl_pmsyrm_thor.rst | 2 +- .../vector/plot_vector_ctrl_syrm_7kw.rst | 2 +- .../vector/sg_execution_times.rst | 14 +- docs/_sources/drive_examples/vhz/index.rst | 4 +- .../vhz/plot_vhz_ctrl_6step_im_2kw.rst | 10 +- .../vhz/plot_vhz_ctrl_im_2kw.rst | 4 +- .../vhz/plot_vhz_ctrl_im_2kw_lc.rst | 2 +- .../drive_examples/vhz/sg_execution_times.rst | 8 +- .../grid_following/plot_gfl_10kva.rst | 24 +- .../grid_following/plot_gfl_dc_bus_10kva.rst | 2 +- .../grid_following/plot_gfl_lcl_10kva.rst | 2 +- .../grid_following/sg_execution_times.rst | 12 +- .../grid_forming/plot_gfm_obs_13kva.rst | 8 +- .../grid_forming/plot_gfm_rfpsc_13kva.rst | 25 +- .../grid_forming/sg_execution_times.rst | 6 +- docs/_sources/sg_execution_times.rst | 56 +- .../motulator/common/control/index.html | 3 +- docs/autoapi/motulator/common/index.html | 3 +- .../autoapi/motulator/common/model/index.html | 3 +- .../autoapi/motulator/common/utils/index.html | 3 +- .../motulator/drive/control/im/index.html | 3 +- .../motulator/drive/control/index.html | 3 +- .../motulator/drive/control/sm/index.html | 3 +- docs/autoapi/motulator/drive/index.html | 3 +- docs/autoapi/motulator/drive/model/index.html | 3 +- docs/autoapi/motulator/drive/utils/index.html | 3 +- .../autoapi/motulator/grid/control/index.html | 15 +- docs/autoapi/motulator/grid/index.html | 3 +- docs/autoapi/motulator/grid/model/index.html | 3 +- docs/autoapi/motulator/grid/utils/index.html | 3 +- docs/autoapi/motulator/index.html | 9 +- docs/contributing.html | 3 +- docs/control/control_system.html | 3 +- docs/control/drive/current_ctrl.html | 3 +- docs/control/drive/index.html | 3 +- docs/control/drive/observers.html | 39 +- docs/control/drive/speed_ctrl.html | 3 +- docs/control/grid/current_ctrl.html | 31 +- docs/control/grid/dc_voltage_ctrl.html | 3 +- docs/control/grid/grid-forming.html | 630 ++++++++++++++++++ docs/control/grid/index.html | 6 +- docs/control/grid/pll.html | 625 +++++++++++++++++ docs/drive_examples/flux_vector/index.html | 3 +- .../flux_vector/plot_flux_vector_im_2kw.html | 5 +- .../plot_flux_vector_pmsm_2kw.html | 5 +- .../plot_flux_vector_pmsyrm_5kw.html | 5 +- .../plot_flux_vector_syrm_7kw.html | 5 +- .../flux_vector/sg_execution_times.html | 13 +- docs/drive_examples/index.html | 7 +- docs/drive_examples/obs_vhz/index.html | 3 +- .../obs_vhz/plot_obs_vhz_ctrl_im_2kw.html | 5 +- .../obs_vhz/plot_obs_vhz_ctrl_pmsm_2kw.html | 5 +- .../plot_obs_vhz_ctrl_pmsm_2kw_two_mass.html | 5 +- .../plot_obs_vhz_ctrl_pmsyrm_thor.html | 5 +- .../obs_vhz/plot_obs_vhz_ctrl_syrm_7kw.html | 5 +- .../obs_vhz/sg_execution_times.html | 15 +- docs/drive_examples/sg_execution_times.html | 3 +- docs/drive_examples/signal_inj/index.html | 3 +- .../signal_inj/plot_signal_inj_pmsm_2kw.html | 5 +- .../signal_inj/plot_signal_inj_syrm_7kw.html | 5 +- .../signal_inj/sg_execution_times.html | 9 +- docs/drive_examples/vector/index.html | 3 +- .../vector/plot_vector_ctrl_im_2kw.html | 5 +- .../plot_vector_ctrl_im_2kw_tq_mode.html | 5 +- .../vector/plot_vector_ctrl_pmsm_2kw.html | 7 +- .../plot_vector_ctrl_pmsm_2kw_diode.html | 5 +- .../vector/plot_vector_ctrl_pmsyrm_thor.html | 5 +- .../vector/plot_vector_ctrl_syrm_7kw.html | 5 +- .../vector/sg_execution_times.html | 17 +- docs/drive_examples/vhz/index.html | 7 +- .../vhz/plot_vhz_ctrl_6step_im_2kw.html | 5 +- .../vhz/plot_vhz_ctrl_im_2kw.html | 5 +- .../vhz/plot_vhz_ctrl_im_2kw_lc.html | 5 +- .../vhz/sg_execution_times.html | 11 +- docs/genindex.html | 3 +- docs/grid_examples/grid_following/index.html | 3 +- .../grid_following/plot_gfl_10kva.html | 9 +- .../grid_following/plot_gfl_dc_bus_10kva.html | 5 +- .../grid_following/plot_gfl_lcl_10kva.html | 5 +- .../grid_following/sg_execution_times.html | 15 +- docs/grid_examples/grid_forming/index.html | 3 +- .../grid_forming/plot_gfm_obs_13kva.html | 11 +- .../grid_forming/plot_gfm_rfpsc_13kva.html | 14 +- .../grid_forming/sg_execution_times.html | 9 +- docs/grid_examples/index.html | 3 +- docs/grid_examples/sg_execution_times.html | 3 +- docs/index.html | 3 +- docs/installation.html | 3 +- docs/model/common.html | 3 +- docs/model/drive/ac_filter.html | 3 +- docs/model/drive/diode_bridge.html | 3 +- docs/model/drive/index.html | 3 +- docs/model/drive/machines.html | 3 +- docs/model/drive/mechanics.html | 3 +- docs/model/grid/filter_and_grid.html | 3 +- docs/model/grid/grid_volt_source.html | 3 +- docs/model/grid/index.html | 3 +- docs/objects.inv | Bin 4749 -> 4781 bytes docs/py-modindex.html | 3 +- docs/search.html | 3 +- docs/searchindex.js | 2 +- docs/sg_execution_times.html | 59 +- docs/usage.html | 3 +- 210 files changed, 2159 insertions(+), 598 deletions(-) create mode 100644 docs/_sources/control/grid/grid-forming.rst create mode 100644 docs/_sources/control/grid/pll.rst create mode 100644 docs/control/grid/grid-forming.html create mode 100644 docs/control/grid/pll.html diff --git a/docs/_downloads/11388a34fc999c2467fbd17556f5ee11/plot_gfm_rfpsc_13kva.ipynb b/docs/_downloads/11388a34fc999c2467fbd17556f5ee11/plot_gfm_rfpsc_13kva.ipynb index cc5bf040..f782747c 100644 --- a/docs/_downloads/11388a34fc999c2467fbd17556f5ee11/plot_gfm_rfpsc_13kva.ipynb +++ b/docs/_downloads/11388a34fc999c2467fbd17556f5ee11/plot_gfm_rfpsc_13kva.ipynb @@ -51,7 +51,7 @@ }, "outputs": [], "source": [ - "# Filter and grid\npar = ACFilterPars(L_fc=.15*base.L, L_g=.74*base.L)\n# par.L_g = 0 # Uncomment this line to simulate a strong grid\nac_filter = model.ACFilter(par)\n# Grid voltage source with constant frequency and voltage magnitude\nac_source = model.ThreePhaseVoltageSource(w_g=base.w, abs_e_g=base.u)\n# Inverter with constant DC voltage\nconverter = model.VoltageSourceConverter(u_dc=650)\n\n# Create system model\nmdl = model.GridConverterSystem(converter, ac_filter, ac_source)" + "# Filter and grid\npar = ACFilterPars(L_fc=.15*base.L, R_fc=.05*base.Z, L_g=.74*base.L)\n# par.L_g = 0 # Uncomment this line to simulate a strong grid\nac_filter = model.ACFilter(par)\n# Grid voltage source with constant frequency and voltage magnitude\nac_source = model.ThreePhaseVoltageSource(w_g=base.w, abs_e_g=base.u)\n# Inverter with constant DC voltage\nconverter = model.VoltageSourceConverter(u_dc=650)\n\n# Create system model\nmdl = model.GridConverterSystem(converter, ac_filter, ac_source)" ] }, { @@ -69,7 +69,7 @@ }, "outputs": [], "source": [ - "# Control configuration parameters\ncfg = control.PowerSynchronizationControlCfg(\n nom_u=base.u, nom_w=base.w, max_i=1.3*base.i, T_s=100e-6, R_a=.2*base.Z)\n\n# Create the control system\nctrl = control.PowerSynchronizationControl(cfg)" + "# Control configuration parameters\ncfg = control.PowerSynchronizationControlCfg(\n nom_u=base.u,\n nom_w=base.w,\n max_i=1.3*base.i,\n R=.05*base.Z,\n R_a=.2*base.Z,\n T_s=100e-6)\n\n# Create the control system\nctrl = control.PowerSynchronizationControl(cfg)" ] }, { diff --git a/docs/_downloads/18fa65c6cad00c3b4897e42fe572224a/grid_examples_jupyter.zip b/docs/_downloads/18fa65c6cad00c3b4897e42fe572224a/grid_examples_jupyter.zip index 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a/docs/_downloads/1901236edeea2d2b57c3aeb0455fdb6c/plot_vhz_ctrl_im_2kw.ipynb +++ b/docs/_downloads/1901236edeea2d2b57c3aeb0455fdb6c/plot_vhz_ctrl_im_2kw.ipynb @@ -4,7 +4,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "\n# 2.2-kW induction motor, diode bridge\n\nA diode bridge, stiff three-phase grid, and a DC link is modeled. The default\nparameters in this example yield open-loop V/Hz control. \n" + "\n# 2.2-kW induction motor, diode bridge\n\nA diode bridge, stiff three-phase grid, and a DC link is modeled. The default\nparameters in this example yield open-loop V/Hz control.\n" ] }, { diff --git a/docs/_downloads/19616cd51216ce55a288135ab2c420f3/drive_examples_jupyter.zip b/docs/_downloads/19616cd51216ce55a288135ab2c420f3/drive_examples_jupyter.zip index 66da09946b2faecac7c777da96e5508567eda73d..2300a8d789a0865905cfd103f56346a48210df09 100644 GIT binary patch delta 524 zcmZY6Pe>GD7{~Gb`Q)P2GP|=Ywu*`%G*F737QECUI~b_ri2ss2DM%S2I)dUs5SENc zLwZ0K1KBQ#P`JCVyQ0Sx@ep3MgW$z*#ZG%sspr*GI55MPci#6M-p_o`dsp*bU)u}1 z6Y}V5#Ib((rVNTY*7|$nrA5;czaxLDXk6k~*Vh2GO3V&LD(R`jcJ!x7w#59>VHq__ zyiIQTX;`8z`>KK#B|aX%4$=pSt9((AOvh41cd=qnNNO#Nod8|5TWaDF$e^%{O*bO+ zfJ&tn+mSKpo|HX+Wy4qDSUtV&ZY*wkJaKh7hC*VmfuH+7;lAQIJT|FTzLbq2UP|J?!wGb12+k(Z zts%HNDI)^GlSy=F2;NV^(GdJF1zSV#R0@3>f)7&iN+7si$tr>1cS=?X1Rv=T2##!^ zOFQs_4q@Qa4cSK&_(>YA8i5zmGFc4xBn|t20)C!BQ`N1*UjNCPM|@_FzhzLXkH!ZX Ylv5%G&P5UC^B9V`Y!iNOPXaRgAGn~Z6951J delta 550 zcmZXRPe@cz6o=oBQ-esun|Y&{{ZS`^2^1G02rh!4h!_z@{u|Q}Ys{MIa5N zQABhwr9~JBQ54P0oq13+5Y-~E7UjZP34zd}HnSsE&;oSRi?mhQ%zWcBqds2_Z z`+Zr{0&3ul^p4E zADY(7FzNO|M4k+gf|I!C#?SbfYp-WW+t7YiU2)on&wfpx^|4p-3 n<_pjdGicS95xjPGn9aVbT+g*Qj*axHvsL{Op(i#}1k-;2Y8taU diff --git a/docs/_downloads/1b80163ae5843d681add6adaacfb4478/plot_flux_vector_pmsyrm_5kw.zip b/docs/_downloads/1b80163ae5843d681add6adaacfb4478/plot_flux_vector_pmsyrm_5kw.zip index 5562b618d8de4625e8d154ef6b43919c430f0d39..4d8db524c6c2ea57c458104bd465ec087b7e86da 100644 GIT binary patch delta 49 zcmaFg#`wOCkvG7bnMH&F1ZsmdHu7dlF}DY6Y_60F)L;e*Og`)62@;qr=^O?CQVR{; delta 49 zcmaFg#`wOCkvG7bnMH&F1WtIXZ{*FCV!q(5zPVB=P=gsLF!`*LCrDtjq;nVmW)TkI diff --git a/docs/_downloads/1bbc064de30c7a3d240a238e31a5d178/plot_vector_ctrl_im_2kw.zip b/docs/_downloads/1bbc064de30c7a3d240a238e31a5d178/plot_vector_ctrl_im_2kw.zip index 3ee54e9a7b0405518ebfaa108cc2db309df16386..8a1eae98794d764a342a747594a0fe072c10366d 100644 GIT binary patch delta 47 xcmZ4Fy~vw4z?+#xgaHJ^gEcntKI37Q4A$8EjfY=?87MHBL)8@|FgZXq2mk^~3f=$! delta 47 xcmZ4Fy~vw4z?+#xgaHInywx}IKI36d^H$&djfY=?87MHBL)8@|FgZXq2mlu<3+?~_ diff --git a/docs/_downloads/2c26aa2b4c9f8800160332fdb6991957/plot_gfm_obs_13kva.zip b/docs/_downloads/2c26aa2b4c9f8800160332fdb6991957/plot_gfm_obs_13kva.zip index 801ef4cc326c8dedb10f9888f82d36f2eadc27b3..dda0d66695ca85b19944c0f5782ffcddb2a53e65 100644 GIT binary patch delta 206 zcmX?Salt}7z?+#xgaHI*25UrqRMD5Z&cVQNeWQ37^W^#b2Ak(Fe_@mjichrFGtx>* zEKb#n(&177f{^%PTSEhbR9&;pYgyBnc^3e6tDB!mDimU1DBP^T*}%4WKEF4!GOE^? zJXF1!F?lh0lQ*zRZ`Kf=%)yJ|gvlqQ&E!#x1sW{@b}EEe(#SPgRz{m`qcn&Q0J1SR A3IG5A delta 215 zcmca$an3>}z?+#xgaHJedaFmCPBI9-#=*bVUc|4zc`ox8M%j?~ zVp~H4gH&BJ9WDhR2#Qa%)icscN-R#*i`u-7HI13~CD0h9zRpVpLJSNAn-w`5*fuZX z_hME?(i)S8p;t2|FD7sDMpo&~TEde#cv0Lk`INMoJc_YEqb0y@h7d~{IVa1@XtS-C G2Jr!_Ogg&& diff --git a/docs/_downloads/353df5bc5a013b3c4d1aa12f9df12b1a/plot_gfm_rfpsc_13kva.py b/docs/_downloads/353df5bc5a013b3c4d1aa12f9df12b1a/plot_gfm_rfpsc_13kva.py index 7f34e924..761d7ec0 100644 --- a/docs/_downloads/353df5bc5a013b3c4d1aa12f9df12b1a/plot_gfm_rfpsc_13kva.py +++ b/docs/_downloads/353df5bc5a013b3c4d1aa12f9df12b1a/plot_gfm_rfpsc_13kva.py @@ -22,7 +22,7 @@ # Configure the system model. # Filter and grid -par = ACFilterPars(L_fc=.15*base.L, L_g=.74*base.L) +par = ACFilterPars(L_fc=.15*base.L, R_fc=.05*base.Z, L_g=.74*base.L) # par.L_g = 0 # Uncomment this line to simulate a strong grid ac_filter = model.ACFilter(par) # Grid voltage source with constant frequency and voltage magnitude @@ -38,7 +38,12 @@ # Control configuration parameters cfg = control.PowerSynchronizationControlCfg( - nom_u=base.u, nom_w=base.w, max_i=1.3*base.i, T_s=100e-6, R_a=.2*base.Z) + nom_u=base.u, + nom_w=base.w, + max_i=1.3*base.i, + R=.05*base.Z, + R_a=.2*base.Z, + T_s=100e-6) # Create the control system ctrl = control.PowerSynchronizationControl(cfg) diff --git a/docs/_downloads/424c2ec214e715617107c56e9505796d/plot_vector_ctrl_im_2kw_tq_mode.zip b/docs/_downloads/424c2ec214e715617107c56e9505796d/plot_vector_ctrl_im_2kw_tq_mode.zip index 912cdf30ca19e3bd9c68f4b8601f1cde5741b631..470bd21a33ba9a9d5ea966a3412449dd9de937cf 100644 GIT binary patch delta 47 xcmZ2$y4I98z?+#xgaHJEf;Bes>TxiO1Z!-z<2WS93>28mEaeLlm>et>1pwwQ3E%(# delta 47 xcmZ2$y4I98z?+#xgaHI%z127J>Txh9daG}?<2WS93>28mEaeLlm>et>1pxZ63hV#? diff --git a/docs/_downloads/4d070dbeed56e687e8fd0f06f72a1494/plot_gfm_obs_13kva.py b/docs/_downloads/4d070dbeed56e687e8fd0f06f72a1494/plot_gfm_obs_13kva.py index 9b680187..897e00cc 100644 --- a/docs/_downloads/4d070dbeed56e687e8fd0f06f72a1494/plot_gfm_obs_13kva.py +++ b/docs/_downloads/4d070dbeed56e687e8fd0f06f72a1494/plot_gfm_obs_13kva.py @@ -25,7 +25,7 @@ # Filter and grid parameters par = ACFilterPars(L_fc=.15*base.L, R_fc=.05*base.Z, L_g=.74*base.L) -# par.L_g = 0 # Uncomment this line to simulate a strong grid +# par.L_g = 0 # Uncomment this line to simulate a strong grid ac_filter = model.ACFilter(par) ac_source = model.ThreePhaseVoltageSource(w_g=base.w, abs_e_g=base.u) # Inverter with constant DC voltage @@ -44,8 +44,8 @@ nom_u=base.u, nom_w=base.w, max_i=1.3*base.i, - T_s=100e-6, - R_a=.2*base.Z) + R_a=.2*base.Z, + T_s=100e-6) # Create the control system ctrl = control.ObserverBasedGridFormingControl(cfg) diff --git a/docs/_downloads/4e23e7d3285bebf832d3d7c99ddcba78/plot_gfl_10kva.zip b/docs/_downloads/4e23e7d3285bebf832d3d7c99ddcba78/plot_gfl_10kva.zip index fb0c90b1dada5ce05f4f3da43d33af5206c3cca9..7c60bda97aea8ea5d0ba1186060c8d8f03caeea1 100644 GIT binary patch delta 174 zcmZoPdub{j;LXe;!T<6vOW+bAx~GWig@^yCB<(aBmYs+&t#UNG}M z1S)A(jo*D+kb&X$=1h(__Q~hPWq@i#m}2rk$|gS&OrJc5mv3@63;*V+0w+0nQB0kj qEoG#DVhGS6i0xn}Ff3_|<(QnuAtn~!&B_K6VgkZOAe|)z;sF34tu8?T diff --git a/docs/_downloads/50647d10d68549b592b9a9cc8f19b847/plot_obs_vhz_ctrl_pmsm_2kw_two_mass.zip b/docs/_downloads/50647d10d68549b592b9a9cc8f19b847/plot_obs_vhz_ctrl_pmsm_2kw_two_mass.zip index 2160b060e11a72179d07098df72809b5c4ba8928..256992ce5eafbf035fd6e80d1c375a72753e0328 100644 GIT binary patch delta 47 xcmeAT>kZ=#@MdNaVE_UDV2zEu#r({H!5W*J`L9Yb0|h2WYXyP?Ca>3u0{{+&3}yfT delta 47 xcmeAT>kZ=#@MdNaVE}=d-s&59i}{&nd#i75=D#Y%3>26gtrZ9on7m#q4gey24Qc=Y diff --git a/docs/_downloads/52056923de1848a7d12a5abf092745c7/plot_gfl_dc_bus_10kva.zip b/docs/_downloads/52056923de1848a7d12a5abf092745c7/plot_gfl_dc_bus_10kva.zip index 197d0aea9b0ea6f8226020cbd34ead734ebf96b8..e30f930a1f0bc38d1fb8bd5cd46158655e79fdb8 100644 GIT binary patch delta 47 xcmexr^3{Ylz?+#xgaHJm1Z!;M72{x@8mzHdjboc2Gf-gieMx7Kz+_da001g{4AB4p delta 47 xcmexr^3{Ylz?+#xgaHKZdaG~b72{xj;H|z{jboc2Gf-gieMx7Kz+_da002H>4b=bu diff --git a/docs/_downloads/5a71f070d06b303d12fe24c7491ee0fd/plot_vector_ctrl_pmsm_2kw_diode.zip b/docs/_downloads/5a71f070d06b303d12fe24c7491ee0fd/plot_vector_ctrl_pmsm_2kw_diode.zip index 7bad2feb75f8a9047e22e0f4bb8d126c14747b83..60f1a0b186db6bf22c80fa484e1297419b0e29c1 100644 GIT binary patch delta 47 xcmaE(_ePI5z?+#xgaHJ&gEcnt3b8W_1#4_pVxP;;3>27rNX!=`Fj+`E3IO;F3RD09 delta 47 xcmaE(_ePI5z?+#xgaHJ?ywx}I3b8ZCdaG|%VxP;;3>27rNX!=`Fj+`E3IGgW3tRvI diff --git a/docs/_downloads/6b891eb683248762e5b2552c3fa1d694/plot_vhz_ctrl_6step_im_2kw.zip b/docs/_downloads/6b891eb683248762e5b2552c3fa1d694/plot_vhz_ctrl_6step_im_2kw.zip index 11a73530e0f71b8322041e0e1f651fa5e7306ad5..c41ed084ac9e3939ceed86da22691346cfd04f2f 100644 GIT binary patch delta 176 zcmcab;@aRG}SEzSS{ delta 184 zcmca;dDBulz?+#xgaHK7ywxL1S6uwT$;rR~!m^WnWfUe_PiIo#njFRD$EYw_k|n_S6xAE+#XD*#BY;PPTth{>Cr&FBNv p!bz?+#xgaHIZf;BesGI27C2WxB=;@m343>27rO4!bz?+#xgaHH+z127JGI27ec&l#~;@m343>27rO429BP|6J?Fj-4F7yuF03y1&! delta 47 xcmexs_ScLzz?+#xgaHKFywx}IvT`tYd#i6229BP|6J?Fj-4F7yu?o43Yo< diff --git a/docs/_downloads/742c9467241727528162f218959331dc/plot_flux_vector_pmsm_2kw.zip b/docs/_downloads/742c9467241727528162f218959331dc/plot_flux_vector_pmsm_2kw.zip index 4161d604f71888b1c8ba84a87956e827550af0d1..2d9714f9f181463438115b3dee57648d1e2acd8d 100644 GIT binary patch delta 47 xcmaE-{Z5-Vz?+#xgaHI{gEcnto?&G!2-etqhqaKO87MG0K-3*1FnN|}2mmla4IKag delta 47 xcmaE-{Z5-Vz?+#xgaHJ$d#i8cJ;Tbp%UgZ(9o9mAW}v|208w|4z~ot?Apk@-4j}*l diff --git a/docs/_downloads/762c2b856ecc40f03ec81b867a35152d/plot_gfm_obs_13kva.ipynb b/docs/_downloads/762c2b856ecc40f03ec81b867a35152d/plot_gfm_obs_13kva.ipynb index 0c2843f9..d40feb6c 100644 --- a/docs/_downloads/762c2b856ecc40f03ec81b867a35152d/plot_gfm_obs_13kva.ipynb +++ b/docs/_downloads/762c2b856ecc40f03ec81b867a35152d/plot_gfm_obs_13kva.ipynb @@ -51,7 +51,7 @@ }, "outputs": [], "source": [ - "# Filter and grid parameters\npar = ACFilterPars(L_fc=.15*base.L, R_fc=.05*base.Z, L_g=.74*base.L)\n# par.L_g = 0 # Uncomment this line to simulate a strong grid\nac_filter = model.ACFilter(par)\nac_source = model.ThreePhaseVoltageSource(w_g=base.w, abs_e_g=base.u)\n# Inverter with constant DC voltage\nconverter = model.VoltageSourceConverter(u_dc=650)\n\n# Create system model\nmdl = model.GridConverterSystem(converter, ac_filter, ac_source)" + "# Filter and grid parameters\npar = ACFilterPars(L_fc=.15*base.L, R_fc=.05*base.Z, L_g=.74*base.L)\n# par.L_g = 0 # Uncomment this line to simulate a strong grid\nac_filter = model.ACFilter(par)\nac_source = model.ThreePhaseVoltageSource(w_g=base.w, abs_e_g=base.u)\n# Inverter with constant DC voltage\nconverter = model.VoltageSourceConverter(u_dc=650)\n\n# Create system model\nmdl = model.GridConverterSystem(converter, ac_filter, ac_source)" ] }, { @@ -69,7 +69,7 @@ }, "outputs": [], "source": [ - "# Set the configuration parameters\ncfg = control.ObserverBasedGridFormingControlCfg(\n L=.35*base.L,\n R=.05*base.Z,\n nom_u=base.u,\n nom_w=base.w,\n max_i=1.3*base.i,\n T_s=100e-6,\n R_a=.2*base.Z)\n\n# Create the control system\nctrl = control.ObserverBasedGridFormingControl(cfg)" + "# Set the configuration parameters\ncfg = control.ObserverBasedGridFormingControlCfg(\n L=.35*base.L,\n R=.05*base.Z,\n nom_u=base.u,\n nom_w=base.w,\n max_i=1.3*base.i,\n R_a=.2*base.Z,\n T_s=100e-6)\n\n# Create the control system\nctrl = control.ObserverBasedGridFormingControl(cfg)" ] }, { diff --git a/docs/_downloads/7caf7c3e3e7e5a0667c115e64a711efe/plot_vector_ctrl_pmsm_2kw.zip b/docs/_downloads/7caf7c3e3e7e5a0667c115e64a711efe/plot_vector_ctrl_pmsm_2kw.zip index 02f0ede85b514344891eff194084dc3c9b9c3203..41e4cc4a7ca05eef08cad4f6f7847d9b96ca81fd 100644 GIT binary patch delta 47 xcmbQHGfjs#z?+#xgaHJ825D^MO=n~N9i*|job4YUGf-f1g{V77VDcf+5CAmg4y^zH delta 47 xcmbQHGfjs#z?+#xgaHKHywx}Irn517daG|PXZy#;3>27LA?gkin0!by1OWUW3&Q{a diff --git a/docs/_downloads/824ec437b909dd213e049bde99b9d868/plot_vhz_ctrl_6step_im_2kw.py b/docs/_downloads/824ec437b909dd213e049bde99b9d868/plot_vhz_ctrl_6step_im_2kw.py index d0cd096e..df6091ba 100644 --- a/docs/_downloads/824ec437b909dd213e049bde99b9d868/plot_vhz_ctrl_6step_im_2kw.py +++ b/docs/_downloads/824ec437b909dd213e049bde99b9d868/plot_vhz_ctrl_6step_im_2kw.py @@ -2,10 +2,10 @@ 2.2-kW induction motor, 6-step mode =================================== -This example simulates V/Hz control of a 2.2-kW induction motor drive. The -six-step overmodulation is enabled, which increases the fundamental voltage as -well as the harmonics. Since the PWM is not synchronized with the stator -frequency, the harmonic content also depends on the ratio between the stator +This example simulates V/Hz control of a 2.2-kW induction motor drive. The +six-step overmodulation is enabled, which increases the fundamental voltage as +well as the harmonics. Since the PWM is not synchronized with the stator +frequency, the harmonic content also depends on the ratio between the stator frequency and the sampling frequency. """ diff --git a/docs/_downloads/86f5cab72ae826f03997079d1a212b48/plot_vhz_ctrl_im_2kw.py b/docs/_downloads/86f5cab72ae826f03997079d1a212b48/plot_vhz_ctrl_im_2kw.py index fbc8a15f..f248e4a2 100644 --- a/docs/_downloads/86f5cab72ae826f03997079d1a212b48/plot_vhz_ctrl_im_2kw.py +++ b/docs/_downloads/86f5cab72ae826f03997079d1a212b48/plot_vhz_ctrl_im_2kw.py @@ -3,7 +3,7 @@ ==================================== A diode bridge, stiff three-phase grid, and a DC link is modeled. The default -parameters in this example yield open-loop V/Hz control. +parameters in this example yield open-loop V/Hz control. """ # %% diff --git a/docs/_downloads/8b8ca3a0a5a7c374a7f4ffc1ecce4ee8/plot_obs_vhz_ctrl_pmsm_2kw.zip b/docs/_downloads/8b8ca3a0a5a7c374a7f4ffc1ecce4ee8/plot_obs_vhz_ctrl_pmsm_2kw.zip index 5f329ec81927c4b8c4f163e379ee920c3ed837be..34d62cab563d1f1f9dd25cdc8f5c69a1aa30e1b6 100644 GIT binary patch delta 47 xcmexv@ZEqnz?+#xgaHI}gEcnt^0G5q1#4`UW#2Bq3>27rUEBjCFj+|=6aWdt3orlx delta 47 xcmexv@ZEqnz?+#xgaHI9ywx}I^0G6xd8==hW#2Bq3>27rUEBjCFj+|=6aXG43_1V+ diff --git a/docs/_downloads/8bf375d70bcac92135f48e77d1408a7f/grid_examples_python.zip b/docs/_downloads/8bf375d70bcac92135f48e77d1408a7f/grid_examples_python.zip index 1a8aef39fdb33e8bd10985880dc1d94baaf2aa7e..270736a3e947a21728cc50bae203378c9a4d7ced 100644 GIT binary patch delta 410 zcmewwI4?*%z?+#xgaHJm25Us_@lfkl=3rn@-YBlbGC7k)l~+MQSs^qpIX^cyHLqlH z0E^`2ewOFV%u|9jHdk`2V*@j^_*FT0X9Bf*Wft&fIFEh#p z#V6Y88EGXY7N_b(>2N6kK}dYDt)YQIs;=4Q|1xRJg42N77ye@uyvN4C0KzJh6U2db zDOfN~o*<>a`KgjH)8wr(x|65L2;nzw@_!k<%?YYs7i{qpYry~j delta 372 zcmbOi^fgdAz?+#xgaHH|c&kTF)2hF%$HBk=!U~gh`K33SF|$lAl9QgC&mub6l0|iL zqLTRJ3BrP#SF;>oX1?pKzB!L$9UGV-&#%hC`xI!%=_G^TYd{08Z4MMqW}f_3DPS|J z^kqi5koaO-Lj!|UT{9gn1t18DPqfuD(n?A!PSuOjoIGDxesZ-c-)3c53ufMjKy%nk zZpN)hmF>Q0TDiaegink_rsXKxkJNcZt zG*b@8WIcYV$*dX@U?Bw!X(m~Ippd3^fHxzP2s0#DP<#aRl@d6xAjFbJ*U1w#4CPT= U3v{sxikj%j4>io$Jk&vg0CUfs;ixQA0v11S5}u;|TN zpkfe~0?JBk{=g;6xS5Ytq830k(zm GAPE4)_a~G9 delta 151 zcmX@&aLhqEz?+#xgaHJyywxMKT!SBUa4|4|u=wON@=_B`uQDoZ=3?B#DOe1Yzy7~; z<~pEq5S9YUOKg76CCkXH5R*6gqPX_t2I;=d$9bL!@uHYDd6&F}Jce$toe*M4 diff --git a/docs/_downloads/a362f5f4469eef05d00e2cd80e2263e6/plot_signal_inj_pmsm_2kw.zip b/docs/_downloads/a362f5f4469eef05d00e2cd80e2263e6/plot_signal_inj_pmsm_2kw.zip index ae0ad485d9c79793518712175e71483605529c63..eda7172c61b34e852c844f235f35846c55f83a93 100644 GIT binary patch delta 47 xcmca27bE8_+dnA|243;-f~47mUR delta 47 xcmcaWjn)AO9Gf-f%t&AH;U~-#GFaS0q4Y>dS diff --git a/docs/_downloads/a684d2b4859af1d04fbd1d02cf9f2d47/plot_gfl_10kva.ipynb b/docs/_downloads/a684d2b4859af1d04fbd1d02cf9f2d47/plot_gfl_10kva.ipynb index 34e04039..853797f6 100644 --- a/docs/_downloads/a684d2b4859af1d04fbd1d02cf9f2d47/plot_gfl_10kva.ipynb +++ b/docs/_downloads/a684d2b4859af1d04fbd1d02cf9f2d47/plot_gfl_10kva.ipynb @@ -51,7 +51,7 @@ }, "outputs": [], "source": [ - "# Filter and grid\npar = ACFilterPars(L_fc=.2*base.L)\nac_filter = model.ACFilter(par)\nac_source = model.ThreePhaseVoltageSource(w_g=base.w, abs_e_g=base.u)\n# Inverter with constant DC voltage\nconverter = model.VoltageSourceConverter(u_dc=650)\n\n# Create system model\nmdl = model.GridConverterSystem(converter, ac_filter, ac_source)\n\n# Uncomment line below to enable the PWM model\n# mdl.pwm = model.CarrierComparison()" + "# Filter and grid\npar = ACFilterPars(L_fc=.2*base.L)\nac_filter = model.ACFilter(par)\nac_source = model.ThreePhaseVoltageSource(w_g=base.w, abs_e_g=base.u)\n# Inverter with constant DC voltage\nconverter = model.VoltageSourceConverter(u_dc=650)\n\n# Create system model\nmdl = model.GridConverterSystem(converter, ac_filter, ac_source)\n# mdl.pwm = model.CarrierComparison() # Uncomment to enable the PWM model" ] }, { diff --git a/docs/_downloads/abf87aac5ab61b081c44e75923d6ef63/plot_gfm_rfpsc_13kva.zip b/docs/_downloads/abf87aac5ab61b081c44e75923d6ef63/plot_gfm_rfpsc_13kva.zip index 081f5d16ad40b0ebe3c5be9a0abfc52455b9ee7d..99f2cc4dfb5df979c5b453677e61d2172039ad02 100644 GIT binary patch delta 383 zcmZoPcw-|KG9as2x=lk zC?vkv*3iHpRo86uDHa7Lp_xDzMaUmEtP@~h0AVSRi)1!Su*b4uIH{N?oH;Ef55r9m zAq+=Bgi%}t6+&?qL=53B&6vEHyvb)-q&KSwxUuu11li>W(Rse KhwY>|hz|gE9Awe} delta 263 zcmaE3&}bkX;LXe;!T=GM|b(l66Fc~sgDdgqn#+TbB zB^IaZmFp5$X delta 47 xcmbPdJa?d4>?>#e?dE@zT3Gf-eMldKy^V6wMtFaRfQ4HW28msp1L}m>j4Q1OWFT3YGu> delta 47 xcmZ4Nx!998z?+#xgaHJKz127JhVd|0c&l$t<5?ii3>28msp1L}m>j4Q1ON+k3!VS~ diff --git a/docs/_downloads/c646707a95e9eded55b2fe6a77579635/plot_obs_vhz_ctrl_pmsyrm_thor.zip b/docs/_downloads/c646707a95e9eded55b2fe6a77579635/plot_obs_vhz_ctrl_pmsyrm_thor.zip index 165b31dd9b925432595730be709007eb5881e898..e31d537c6e3e1ad363517eee83c985e271d29d87 100644 GIT binary patch delta 47 xcmaDD_b`q(z?+#xgaHHsgEcnt1_(0e25W3i5R8yz1`15xt>+CAn9Qjk0RSe#3{?OC delta 47 xcmaDD_b`q(z?+#xgaHI*d#i8c4G?7B?ybH#K`=s=87MG$x1KjhU^1tE1OPg?4OsvH diff --git a/docs/_downloads/ca790e45bb116fa02f80f2d12b72410d/plot_flux_vector_syrm_7kw.zip b/docs/_downloads/ca790e45bb116fa02f80f2d12b72410d/plot_flux_vector_syrm_7kw.zip index 7bc78d8aa5024db3447ad5febcc8ef419c2aede4..a6913729dc81ff2e0e789dfc6d3dc5883602a30c 100644 GIT binary patch delta 47 xcmbP}F(-pJz?+#xgaHIfgEcntwhA-X25W4dD*Q~I87MHh&e$C!F!_XW2mmeV4afih delta 47 xcmbP}F(-pJz?+#xgaHH&daG~bZ53ud;jO-Ts_-*;W}v|2I%9W`z~mFgApk;L4$=Ss diff --git a/docs/_downloads/ce870960718a5060422b2fb3301e5579/plot_obs_vhz_ctrl_syrm_7kw.zip b/docs/_downloads/ce870960718a5060422b2fb3301e5579/plot_obs_vhz_ctrl_syrm_7kw.zip index 3cc1f5eeb5b55201353e30892fa784817f6b8e58..3cd8e7d938a90b2e242ab2a0fac39f741312d308 100644 GIT binary patch delta 47 xcmX?|d^(voz?+#xgaHH`f;BesJ{My457yZHUC2<587MGW+t33fFuBYy6aX$S47&gT delta 47 xcmX?|d^(voz?+#xgaHJ)z127JJ{Mx1>8-x`yO5zAGf-f%wxI_|U~-vZC;&oK4Z#2a diff --git a/docs/_downloads/e4f25fcaeb625e6a8f40d3c21f859314/drive_examples_python.zip b/docs/_downloads/e4f25fcaeb625e6a8f40d3c21f859314/drive_examples_python.zip index 98ca9d52db843551262c186249bd63fb693e8726..f4b32114259ced1873d17fe5710c56b905ebf2db 100644 GIT binary patch delta 515 zcmZXRODIH97{_mYX6{@wx5*e`#9(zx7V=n_6v`{i%yp3|VP!{BHsrM;WidW!P-;Y0 z78)BxHbypBcqCcQd&natvG9Gfk(~O@`E}0uI{p6V;}+g-L3X!7npNO2q$zvyoem~7 zB@ok%*vqVq{7KVB1AE@gt*X$<-9k z*;?PBP7(yJA_41?Ry<0Wv82L=*C&(M5p>~ZumvB3?Sbl36fODGlmBq#m`fJ~t~~Cs z;qHzdb1n|BxWy zTxDx?B%JaeXG!(7zpBrQ7gUdAHkJK2h_d4;LyD0u%!BL%1JKdAJp cN@I87;-#lJ4dVa+ delta 47 ycmdmPvfYF?z?+#xgaHJ0d8=>aeb3H(&|7`;fA-me%s_$3b0u9t0+XLg1_1y=EDq}c diff --git a/docs/_downloads/f1af7679d3b257951ac507da64a82b0e/plot_vector_ctrl_syrm_7kw.zip b/docs/_downloads/f1af7679d3b257951ac507da64a82b0e/plot_vector_ctrl_syrm_7kw.zip index 3b4b53e62de08e7867a6422c363f1d018a5f5867..e506af481659f55021471c9328bad9a86883a2e5 100644 GIT binary patch delta 47 xcmX@7cTSHtz?+#xgaHJ82Wf2NWnpLL4%XN#!akFq87MG$gP1!=VDc}q5C9i943Pi; delta 47 xcmX@7cTSHtz?+#xgaHIRz127JvamCUd8=<0VV}v*3>28WLChT_F!`5Q2mt*i3!wl2 diff --git a/docs/_downloads/fcedaf8dcbc7bd92d429227a15419813/plot_vhz_ctrl_6step_im_2kw.ipynb b/docs/_downloads/fcedaf8dcbc7bd92d429227a15419813/plot_vhz_ctrl_6step_im_2kw.ipynb index e893fbd5..a869c4cc 100644 --- a/docs/_downloads/fcedaf8dcbc7bd92d429227a15419813/plot_vhz_ctrl_6step_im_2kw.ipynb +++ b/docs/_downloads/fcedaf8dcbc7bd92d429227a15419813/plot_vhz_ctrl_6step_im_2kw.ipynb @@ -4,7 +4,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "\n# 2.2-kW induction motor, 6-step mode\n\nThis example simulates V/Hz control of a 2.2-kW induction motor drive. The \nsix-step overmodulation is enabled, which increases the fundamental voltage as \nwell as the harmonics. Since the PWM is not synchronized with the stator \nfrequency, the harmonic content also depends on the ratio between the stator \nfrequency and the sampling frequency.\n" + "\n# 2.2-kW induction motor, 6-step mode\n\nThis example simulates V/Hz control of a 2.2-kW induction motor drive. The\nsix-step overmodulation is enabled, which increases the fundamental voltage as\nwell as the harmonics. Since the PWM is not synchronized with the stator\nfrequency, the harmonic content also depends on the ratio between the stator\nfrequency and the sampling frequency.\n" ] }, { diff --git a/docs/_images/sphx_glr_plot_gfm_obs_13kva_001.png b/docs/_images/sphx_glr_plot_gfm_obs_13kva_001.png index 7ad7236e5cb2dce9e5ea80f6dee943d9861d1282..a0b69683e4e82b78124a76d97fe5c6acb60f53c7 100644 GIT binary patch delta 166156 zcmV)}KzqOMlN|7q9DsxYgaU*Ev;>}dfAr?l@XycrtA877UounpxPCSsFqfi~iPuJ| zPWsnorQRBHvou~J$qdg4f_RX5B_#>>);00?2cBZ({bB$z5!cTM_iA&UO=tRUPk~t*I@zF%~<))6m zSO?%Q+5tFw);#PJV_%)z>MM>@e+Yl*x-S`HJ6GNl`<<^Up8){mC<#+ip`!m-Be>V3 zcL){|DvoDsqu$@SO%DM(v~NA8RK`#zow(cF359&TlZ2a0mSQ|6&&1j4H4tD?=tKkc z@%5f6EupCKt5!BlkV<=A7W3xA-PQ&LDy$AkU3e-(h<1f=#U z_3v8qDWoW0asePmk$y@DLDO;4Rx}PBg<B~JzAoEpl;Q^sKuf4b-Q2nw9~5rH1J;fph))@-mB-<_ICKg@v$AU89QD)qed_Z# zw0;YK_N5P&e?`2UiYh{9k2|yQQPnG+}E&+ zcpH9c9E$$IgJSK?x_{l%BpXj^JchV&9Kt_=;jJbR(=4Y^p^E^zsFudOm$)W{jzL4! 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diff --git a/docs/_modules/motulator/common/control/_control.html b/docs/_modules/motulator/common/control/_control.html index fb05e02d..e9974177 100644 --- a/docs/_modules/motulator/common/control/_control.html +++ b/docs/_modules/motulator/common/control/_control.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/common/model/_converter.html b/docs/_modules/motulator/common/model/_converter.html index 13892758..e2d8bde0 100644 --- a/docs/_modules/motulator/common/model/_converter.html +++ b/docs/_modules/motulator/common/model/_converter.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/common/model/_simulation.html b/docs/_modules/motulator/common/model/_simulation.html index c4ea1203..f60a69a2 100644 --- a/docs/_modules/motulator/common/model/_simulation.html +++ b/docs/_modules/motulator/common/model/_simulation.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/common/utils/_utils.html b/docs/_modules/motulator/common/utils/_utils.html index 8f6f8629..d5402b13 100644 --- a/docs/_modules/motulator/common/utils/_utils.html +++ b/docs/_modules/motulator/common/utils/_utils.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/control/_common.html b/docs/_modules/motulator/drive/control/_common.html index 2a3e8eb6..68e19aa4 100644 --- a/docs/_modules/motulator/drive/control/_common.html +++ b/docs/_modules/motulator/drive/control/_common.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/control/im/_common.html b/docs/_modules/motulator/drive/control/im/_common.html index adc27b8f..18b1886a 100644 --- a/docs/_modules/motulator/drive/control/im/_common.html +++ b/docs/_modules/motulator/drive/control/im/_common.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/control/im/_current_vector.html b/docs/_modules/motulator/drive/control/im/_current_vector.html index 1bc5dcd1..de9bb639 100644 --- a/docs/_modules/motulator/drive/control/im/_current_vector.html +++ b/docs/_modules/motulator/drive/control/im/_current_vector.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/control/im/_flux_vector.html b/docs/_modules/motulator/drive/control/im/_flux_vector.html index 06e9d484..c1735d35 100644 --- a/docs/_modules/motulator/drive/control/im/_flux_vector.html +++ b/docs/_modules/motulator/drive/control/im/_flux_vector.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/control/im/_obs_vhz.html b/docs/_modules/motulator/drive/control/im/_obs_vhz.html index 91e6c528..a362fc6b 100644 --- a/docs/_modules/motulator/drive/control/im/_obs_vhz.html +++ b/docs/_modules/motulator/drive/control/im/_obs_vhz.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/control/im/_vhz.html b/docs/_modules/motulator/drive/control/im/_vhz.html index 583f1a26..22e7df83 100644 --- a/docs/_modules/motulator/drive/control/im/_vhz.html +++ b/docs/_modules/motulator/drive/control/im/_vhz.html @@ -226,7 +226,8 @@
  • Grid Converters
    • @@ -419,7 +420,7 @@

    Source code for motulator.drive.control.im._vhz

    < """ V/Hz control with the stator current feedback. - The method is similar to [#Hin2022]_. Open-loop V/Hz control can be + The method is similar to [#Hin2022]_. Open-loop V/Hz control can be obtained as a special case by choosing:: R_s, R_R = 0, 0 @@ -428,8 +429,8 @@

    Source code for motulator.drive.control.im._vhz

    < References ---------- .. [#Hin2022] Hinkkanen, Tiitinen, Mölsä, Harnefors, "On the stability of - volts-per-hertz control for induction motors," IEEE J. Emerg. Sel. - Topics Power Electron., 2022, + volts-per-hertz control for induction motors," IEEE J. Emerg. Sel. + Topics Power Electron., 2022, https://doi.org/10.1109/JESTPE.2021.3060583 """ diff --git a/docs/_modules/motulator/drive/control/sm/_common.html b/docs/_modules/motulator/drive/control/sm/_common.html index c7c02dfd..d8062095 100644 --- a/docs/_modules/motulator/drive/control/sm/_common.html +++ b/docs/_modules/motulator/drive/control/sm/_common.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/control/sm/_current_vector.html b/docs/_modules/motulator/drive/control/sm/_current_vector.html index bd98a3bc..eb5a5290 100644 --- a/docs/_modules/motulator/drive/control/sm/_current_vector.html +++ b/docs/_modules/motulator/drive/control/sm/_current_vector.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/control/sm/_flux_vector.html b/docs/_modules/motulator/drive/control/sm/_flux_vector.html index b6e1a303..d67d17bb 100644 --- a/docs/_modules/motulator/drive/control/sm/_flux_vector.html +++ b/docs/_modules/motulator/drive/control/sm/_flux_vector.html @@ -226,7 +226,8 @@
  • Grid Converters
    • @@ -392,11 +393,11 @@

    Source code for motulator.drive.control.sm._flux_vector

    """ Flux-vector control of synchronous machine drives. - This class implements a variant of flux-vector control [#Pel2009]_. Rotor - coordinates as well as decoupling between the stator flux and torque - channels are used according to [#Awa2019b]_. Here, the stator flux - magnitude and the electromagnetic torque are selected as controllable - variables, and proportional controllers are used for simplicity + This class implements a variant of flux-vector control [#Pel2009]_. Rotor + coordinates as well as decoupling between the stator flux and torque + channels are used according to [#Awa2019b]_. Here, the stator flux + magnitude and the electromagnetic torque are selected as controllable + variables, and proportional controllers are used for simplicity [#Tii2024]_. The magnetic saturation is not considered in this implementation. @@ -413,7 +414,7 @@

    Source code for motulator.drive.control.sm._flux_vector

    alpha_o : float, optional Observer bandwidth (rad/s). The default is 2*pi*100. J : float, optional - Moment of inertia (kgm²). Needed only for the speed controller. + Moment of inertia (kgm²). Needed only for the speed controller. T_s : float Sampling period (s). The default is 250e-6. sensorless : bool, optional @@ -421,18 +422,18 @@

    Source code for motulator.drive.control.sm._flux_vector

    References ---------- - .. [#Pel2009] Pellegrino, Armando, Guglielmi, “Direct flux field-oriented - control of IPM drives with variable DC link in the field-weakening - region,” IEEE Trans. Ind. Appl., 2009, + .. [#Pel2009] Pellegrino, Armando, Guglielmi, “Direct flux field-oriented + control of IPM drives with variable DC link in the field-weakening + region,” IEEE Trans. Ind. Appl., 2009, https://doi.org/10.1109/TIA.2009.2027167 - .. [#Awa2019b] Awan, Hinkkanen, Bojoi, Pellegrino, "Stator-flux-oriented - control of synchronous motors: A systematic design procedure," IEEE + .. [#Awa2019b] Awan, Hinkkanen, Bojoi, Pellegrino, "Stator-flux-oriented + control of synchronous motors: A systematic design procedure," IEEE Trans. Ind. Appl., 2019, https://doi.org/10.1109/TIA.2019.2927316 - .. [#Tii2024] Tiitinen, Hinkkanen, Harnefors, "Design framework for - sensorless control of synchronous machine drives," IEEE Trans. Ind. - Electron., 2024, https://doi.org/10.1109/TIE.2024.3429650 + .. [#Tii2024] Tiitinen, Hinkkanen, Harnefors, "Design framework for + sensorless control of synchronous machine drives," IEEE Trans. Ind. + Electron., 2024, https://doi.org/10.1109/TIE.2024.3429650 """ @@ -504,7 +505,7 @@

    Source code for motulator.drive.control.sm._flux_vector

    par : SynchronousMachinePars Machine model parameters. max_i_s : float - Maximum stator current (A). + Maximum stator current (A). min_psi_s : float, optional Minimum stator flux (Vs). The default is `par.psi_f`. max_psi_s : float, optional diff --git a/docs/_modules/motulator/drive/control/sm/_obs_vhz.html b/docs/_modules/motulator/drive/control/sm/_obs_vhz.html index d374d132..1b492828 100644 --- a/docs/_modules/motulator/drive/control/sm/_obs_vhz.html +++ b/docs/_modules/motulator/drive/control/sm/_obs_vhz.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/control/sm/_signal_inj.html b/docs/_modules/motulator/drive/control/sm/_signal_inj.html index 1bd8e375..6205dcf8 100644 --- a/docs/_modules/motulator/drive/control/sm/_signal_inj.html +++ b/docs/_modules/motulator/drive/control/sm/_signal_inj.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/control/sm/_torque.html b/docs/_modules/motulator/drive/control/sm/_torque.html index 04095398..ab670ee0 100644 --- a/docs/_modules/motulator/drive/control/sm/_torque.html +++ b/docs/_modules/motulator/drive/control/sm/_torque.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/model/_drive.html b/docs/_modules/motulator/drive/model/_drive.html index f8f26077..d27fd8ff 100644 --- a/docs/_modules/motulator/drive/model/_drive.html +++ b/docs/_modules/motulator/drive/model/_drive.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/model/_lc_filter.html b/docs/_modules/motulator/drive/model/_lc_filter.html index ca6cdad9..6063eee7 100644 --- a/docs/_modules/motulator/drive/model/_lc_filter.html +++ b/docs/_modules/motulator/drive/model/_lc_filter.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/model/_machine.html b/docs/_modules/motulator/drive/model/_machine.html index 62aaace7..ba3611c0 100644 --- a/docs/_modules/motulator/drive/model/_machine.html +++ b/docs/_modules/motulator/drive/model/_machine.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/model/_mechanics.html b/docs/_modules/motulator/drive/model/_mechanics.html index a1b25462..c648564f 100644 --- a/docs/_modules/motulator/drive/model/_mechanics.html +++ b/docs/_modules/motulator/drive/model/_mechanics.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/utils/_flux_maps.html b/docs/_modules/motulator/drive/utils/_flux_maps.html index e3b78075..146519e9 100644 --- a/docs/_modules/motulator/drive/utils/_flux_maps.html +++ b/docs/_modules/motulator/drive/utils/_flux_maps.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/utils/_helpers.html b/docs/_modules/motulator/drive/utils/_helpers.html index acbb4c82..c87c6bb0 100644 --- a/docs/_modules/motulator/drive/utils/_helpers.html +++ b/docs/_modules/motulator/drive/utils/_helpers.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/drive/utils/_plots.html b/docs/_modules/motulator/drive/utils/_plots.html index 108b65ad..e31acf14 100644 --- a/docs/_modules/motulator/drive/utils/_plots.html +++ b/docs/_modules/motulator/drive/utils/_plots.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/grid/control/_common.html b/docs/_modules/motulator/grid/control/_common.html index 73dd1437..d563ee5e 100644 --- a/docs/_modules/motulator/grid/control/_common.html +++ b/docs/_modules/motulator/grid/control/_common.html @@ -226,7 +226,8 @@
  • Grid Converters
    • @@ -391,18 +392,18 @@

    Source code for motulator.grid.control._common

    """ Phase-locked loop including the voltage-magnitude filtering. - This class provides a simple frequency-tracking phase-locked loop. The - magnitude of the measured PCC voltage is also filtered. + This class provides a simple frequency-tracking phase-locked loop. The + magnitude of the measured PCC voltage is also filtered. Parameters ---------- alpha_pll : float Frequency-tracking bandwidth (rad/s). abs_u_g0 : float - Initial value for the grid voltage estimate (V). + Initial value for the grid voltage estimate (V). w_g0 : float - Initial value for the grid angular frequency estimate (rad/s). - + Initial value for the grid angular frequency estimate (rad/s). + """ def __init__(self, alpha_pll, abs_u_g0, w_g0, theta_c0=0): @@ -439,19 +440,19 @@

    Source code for motulator.grid.control._common

    """ DC-bus voltage PI controller. - This controller regulates the energy stored in the DC-bus capacitor (scaled - square of the DC-bus voltage) in order to have a linear closed-loop system - [#Hur2001]_. + This controller regulates the energy stored in the DC-bus capacitor (scaled + square of the DC-bus voltage) in order to have a linear closed-loop system + [#Hur2001]_. Parameters ---------- C_dc : float DC-bus capacitance (F). alpha_dc : float - Approximate closed-loop bandwidth (rad/s). + Approximate closed-loop bandwidth (rad/s). max_p : float, optional Limit for the maximum converter power (W). The default is `inf`. - + References ---------- .. [#Hur2001] Hur, Jung, Nam, "A fast dynamic DC-link power-balancing @@ -477,10 +478,10 @@

    Source code for motulator.grid.control._common

    [docs]class GridConverterControlSystem(ControlSystem, ABC): """ Base class for control of grid-connected converters. - + This base class provides typical functionalities for control of grid-connected converters. This can be used both in power control and - DC-bus voltage control modes. + DC-bus voltage control modes. Parameters ---------- @@ -494,7 +495,7 @@

    Source code for motulator.grid.control._common

    v : float | callable Converter output voltage reference (V). Can be given either as - a constant or a function of time (s). + a constant or a function of time (s). p_g : callable Active power reference (W) as a function of time (s). This signal is needed in power control mode. @@ -519,7 +520,7 @@

    Source code for motulator.grid.control._common

    [docs] def get_electrical_measurements(self, fbk, mdl): """ Measure the currents and voltages. - + Parameters ---------- fbk : SimpleNamespace @@ -539,15 +540,11 @@

    Source code for motulator.grid.control._common

    u_cs : complex Realized converter output voltage (V) in stationary coordinates. This signal is obtained from the PWM. - u_gs : complex - PCC voltage (V) in stationary coordinates. """ fbk.u_dc = mdl.converter.meas_dc_voltage() fbk.i_cs = abc2complex(mdl.ac_filter.meas_currents()) fbk.u_cs = self.pwm.get_realized_voltage() - fbk.u_gs = abc2complex(mdl.ac_filter.meas_pcc_voltages()) - # fbk.u_gs = abc2complex(mdl.ac_filter.meas_capacitor_voltages()) return fbk

    @@ -572,14 +569,14 @@

    Source code for motulator.grid.control._common

    Returns ------- ref : SimpleNamespace - Reference signals, containing the following fields: - + Reference signals, containing some of the following fields: + u_dc : float DC-bus voltage reference (V). p_g : float Active power reference (W). q_g : float - Reactive power reference (VAr). + Reactive power reference (VAr). """ if self.dc_bus_voltage_ctrl: @@ -588,12 +585,12 @@

    Source code for motulator.grid.control._common

    ref.p_g = self.dc_bus_voltage_ctrl.output(ref.u_dc, fbk.u_dc) else: # Power control mode - #ref.u_dc = None ref.p_g = self.ref.p_g(ref.t) - # Reactive power reference - ref.q_g = self.ref.q_g(ref.t) if callable( - self.ref.q_g) else self.ref.q_g + # Reactive power reference (if exists) + if hasattr(self.ref, 'q_g'): + ref.q_g = self.ref.q_g(ref.t) if callable( + self.ref.q_g) else self.ref.q_g return ref

    diff --git a/docs/_modules/motulator/grid/control/_grid_following.html b/docs/_modules/motulator/grid/control/_grid_following.html index 6e3f80c9..e1c88068 100644 --- a/docs/_modules/motulator/grid/control/_grid_following.html +++ b/docs/_modules/motulator/grid/control/_grid_following.html @@ -226,7 +226,8 @@
  • Grid Converters
    • @@ -384,6 +385,7 @@

    Source code for motulator.grid.control._grid_following

    from motulator.common.control import ComplexPIController from motulator.grid.control._common import ( CurrentLimiter, GridConverterControlSystem, PLL) +from motulator.common.utils import abc2complex # %% @@ -449,6 +451,8 @@

    Source code for motulator.grid.control._grid_following

    [docs] def get_feedback_signals(self, mdl): fbk = super().get_feedback_signals(mdl) + fbk.u_gs = abc2complex(mdl.ac_filter.meas_pcc_voltages()) + # fbk.u_gs = abc2complex(mdl.ac_filter.meas_capacitor_voltages()) fbk = self.pll.output(fbk) s_g = 1.5*fbk.u_c*np.conj(fbk.i_c) fbk.p_g = s_g.real diff --git a/docs/_modules/motulator/grid/control/_observer_gfm.html b/docs/_modules/motulator/grid/control/_observer_gfm.html index 30e4d07d..e2c6a08e 100644 --- a/docs/_modules/motulator/grid/control/_observer_gfm.html +++ b/docs/_modules/motulator/grid/control/_observer_gfm.html @@ -226,7 +226,8 @@
  • Grid Converters
    • @@ -382,7 +383,8 @@

    Source code for motulator.grid.control._observer_gfm

    import numpy as np from motulator.common.utils import wrap -from motulator.grid.control._common import GridConverterControlSystem +from motulator.grid.control._common import ( + GridConverterControlSystem, CurrentLimiter) # %% @@ -405,12 +407,14 @@

    Source code for motulator.grid.control._observer_gfm

    Total series resistance (Ω). The default is 0. R_a : float, optional Active resistance (Ω). The default is 0.25*nom_u/max_i. - T_s : float, optional - Sampling period (s). The default is 100e-6. + k_v : float, optional + Voltage control gain. The default is `alpha_o/nom_w`. alpha_c : float, optional Current control bandwidth (rad/s). The default is 2*pi*400. alpha_o : float, optional Observer gain (rad/s). The default is 2*pi*50. + T_s : float, optional + Sampling period (s). The default is 100e-6. """ L: float @@ -419,13 +423,16 @@

    Source code for motulator.grid.control._observer_gfm

    max_i: float R: float = 0 R_a: float = None - T_s: float = 100e-6 + k_v: float = None alpha_c: float = 2*np.pi*400 alpha_o: float = 2*np.pi*50 + T_s: float = 100e-6 def __post_init__(self): if self.R_a is None: self.R_a = .25*self.nom_u/self.max_i + if self.k_v is None: + self.k_v = self.alpha_o/self.nom_w self.k_c = self.alpha_c*self.L # Current control gain
    @@ -440,7 +447,7 @@

    Source code for motulator.grid.control._observer_gfm

    Parameters ---------- cfg : ObserverBasedGridFormingControlCfg - Controller configuration parameters. + Control system configuration parameters. Notes ----- @@ -460,9 +467,8 @@

    Source code for motulator.grid.control._observer_gfm

    def __init__(self, cfg): super().__init__(cfg.T_s) self.cfg = cfg - self.observer = DisturbanceObserver( - cfg.nom_w, cfg.L, cfg.alpha_o, cfg.nom_u) - self.ref.q_g = 0 + self.observer = DisturbanceObserver(cfg) + self.current_limiter = CurrentLimiter(cfg.max_i)
    [docs] def get_feedback_signals(self, mdl): """Get the feedback signals.""" @@ -485,21 +491,19 @@

    Source code for motulator.grid.control._observer_gfm

    self.ref.v_c) else self.ref.v_c # Complex gains for grid-forming mode - abs_k_p = cfg.R_a/(1.5*ref.v_c) - abs_v_c = np.abs(fbk.v_c) - k_p = abs_k_p*fbk.v_c/abs_v_c if abs_v_c > 0 else 0 - k_v = (1 - 1j)*fbk.v_c/abs_v_c if abs_v_c > 0 else 0 + exp_j_theta = fbk.v_c/np.abs(fbk.v_c) if np.abs(fbk.v_c) > 0 else 1 + k_p = cfg.R_a/(1.5*ref.v_c)*exp_j_theta + k_v = (1 - 1j*cfg.k_v)*exp_j_theta # Feedback correction for grid-forming mode - fbk.e_c = k_p*(ref.p_g - fbk.p_g) + k_v*(ref.v_c - abs_v_c) + fbk.e_c = k_p*(ref.p_g - fbk.p_g) + k_v*(ref.v_c - np.abs(fbk.v_c)) - # Current limitation + # Transparent current limitation ref.i_c = fbk.i_c + fbk.e_c/cfg.k_c - if np.abs(ref.i_c) > cfg.max_i: - ref.i_c_lim = ref.i_c/np.abs(ref.i_c)*cfg.max_i - fbk.e_c = cfg.k_c*(ref.i_c_lim - fbk.i_c) + ref.i_c = self.current_limiter(ref.i_c) + fbk.e_c = cfg.k_c*(ref.i_c - fbk.i_c) - # Voltage reference + # Voltage reference (with optional resistive voltage drop compensation) ref.u_c = fbk.e_c + fbk.v_c + cfg.R*fbk.i_c ref.u_cs = np.exp(1j*fbk.theta_c)*ref.u_c @@ -524,42 +528,32 @@

    Source code for motulator.grid.control._observer_gfm

    Parameters ---------- - w_g : float - Nominal grid angular frequency (rad/s). - L : float - Estimate of total inductance (H) between converter and grid. - alpha_o : float - Observer gain (rad/s). - v_c0 : float - Initial value of converter voltage state (V). + cfg : ObserverBasedGridFormingControlCfg + Control system configuration parameters. """ - def __init__(self, w_g, L, alpha_o, v_c0): - self.w_g = w_g - self.w_c = w_g - self.L = L - self.k_o = alpha_o - 1j*w_g - # Initial states - self.v_cp = v_c0 + def __init__(self, cfg): + self.alpha_o = cfg.alpha_o + self.L = cfg.L + self.w_g = cfg.nom_w + self.u_gp = cfg.nom_u self.theta_c = 0 def output(self, fbk): """Compute the estimates.""" - # Quasi-static converter voltage - fbk.v_c = self.v_cp - self.k_o*self.L*fbk.i_c - # Grid voltage - fbk.u_g = fbk.v_c - 1j*self.w_g*self.L*fbk.i_c - # Active and reactive power - fbk.p_g = 1.5*np.real(fbk.v_c*np.conj(fbk.i_c)) - fbk.q_g = 1.5*np.imag(fbk.u_g*np.conj(fbk.i_c)) + # Estimates for the quasi-static converter voltage and the grid voltage + fbk.v_c = self.u_gp - (self.alpha_o - 1j*self.w_g)*self.L*fbk.i_c + fbk.u_g = self.u_gp - self.alpha_o*self.L*fbk.i_c + # Active and reactive powers + s_g = 1.5*fbk.u_g*np.conj(fbk.i_c) + fbk.p_g, fbk.q_g = s_g.real, s_g.imag return fbk def update(self, fbk, ref): - """Update the observer states.""" - self.v_cp += ref.T_s*(self.k_o + 1j*self.w_c)*fbk.e_c + 1j*( - self.w_g - self.w_c)*self.v_cp - self.theta_c += ref.T_s*self.w_c + """Update the states.""" + self.u_gp += ref.T_s*self.alpha_o*fbk.e_c + self.theta_c += ref.T_s*self.w_g self.theta_c = wrap(self.theta_c)
    diff --git a/docs/_modules/motulator/grid/control/_power_synchronization.html b/docs/_modules/motulator/grid/control/_power_synchronization.html index 1afb9363..db4bc583 100644 --- a/docs/_modules/motulator/grid/control/_power_synchronization.html +++ b/docs/_modules/motulator/grid/control/_power_synchronization.html @@ -226,7 +226,8 @@
  • Grid Converters
    • @@ -400,20 +401,23 @@

    Source code for motulator.grid.control._power_synchronization

    Nominal grid angular frequency (rad/s). max_i : float Maximum current (A), peak value. + R : float, optional + Total series resistance (Ω). The default is 0. R_a : float, optional Active resistance (Ω). The default is 0.25*nom_u/max_i. - T_s : float, optional - Sampling period (s). The default is 100e-6. w_b : float, optional Low-pass filter bandwidth (rad/s). The default is 2*pi*5. + T_s : float, optional + Sampling period (s). The default is 100e-6. """ nom_u: float nom_w: float max_i: float + R: float = 0 R_a: float = None - T_s: float = 100e-6 w_b: float = 2*np.pi*5 + T_s: float = 100e-6 def __post_init__(self): if self.R_a is None: @@ -437,8 +441,8 @@

    Source code for motulator.grid.control._power_synchronization

    References ---------- .. [#Har2020] Harnefors, Rahman, Hinkkanen, Routimo, "Reference-feedforward - power-synchronization control," IEEE Trans. Power Electron., 2020, - https://doi.org/10.1109/TPEL.2020.2970991 + power-synchronization control," IEEE Trans. Power Electron., 2020, + https://doi.org/10.1109/TPEL.2020.2970991 """ @@ -446,7 +450,6 @@

    Source code for motulator.grid.control._power_synchronization

    super().__init__(cfg.T_s) self.cfg = cfg self.current_limiter = CurrentLimiter(cfg.max_i) - self.ref.q_g = 0 self.theta_c = 0 self.i_c_flt = 0j @@ -459,7 +462,8 @@

    Source code for motulator.grid.control._power_synchronization

    fbk.i_c = np.exp(-1j*fbk.theta_c)*fbk.i_cs fbk.u_c = np.exp(-1j*fbk.theta_c)*fbk.u_cs # Active power - fbk.p_c = 1.5*np.real(fbk.u_c*np.conj(fbk.i_c)) + p_loss = 1.5*self.cfg.R*np.abs(fbk.i_c)**2 + fbk.p_g = 1.5*np.real(fbk.u_c*np.conj(fbk.i_c)) - p_loss return fbk
    [docs] def output(self, fbk): @@ -473,7 +477,7 @@

    Source code for motulator.grid.control._power_synchronization

    self.ref.v_c) else self.ref.v_c # Calculation of power droop - fbk.w_c = cfg.nom_w + cfg.k_p_psc*(ref.p_g - fbk.p_c) + fbk.w_c = cfg.nom_w + cfg.k_p_psc*(ref.p_g - fbk.p_g) # Optionally, use of reference feedforward for d-axis current ref.i_c = ref.p_g/(1.5*ref.v_c) + 1j*fbk.i_c_flt.imag @@ -483,7 +487,7 @@

    Source code for motulator.grid.control._power_synchronization

    ref.i_c = self.current_limiter(ref.i_c) # Calculation of converter voltage output reference - ref.u_c = ref.v_c + cfg.R_a*(ref.i_c - fbk.i_c) + ref.u_c = ref.v_c + cfg.R_a*(ref.i_c - fbk.i_c) + cfg.R*fbk.i_c ref.u_cs = np.exp(1j*fbk.theta_c)*ref.u_c # Duty ratios for PWM diff --git a/docs/_modules/motulator/grid/model/_ac_filter.html b/docs/_modules/motulator/grid/model/_ac_filter.html index d82e73e0..2679fe2b 100644 --- a/docs/_modules/motulator/grid/model/_ac_filter.html +++ b/docs/_modules/motulator/grid/model/_ac_filter.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/grid/model/_converter_system.html b/docs/_modules/motulator/grid/model/_converter_system.html index 73c42f5f..2f3a8587 100644 --- a/docs/_modules/motulator/grid/model/_converter_system.html +++ b/docs/_modules/motulator/grid/model/_converter_system.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/grid/model/_voltage_source.html b/docs/_modules/motulator/grid/model/_voltage_source.html index e89580f7..14a39544 100644 --- a/docs/_modules/motulator/grid/model/_voltage_source.html +++ b/docs/_modules/motulator/grid/model/_voltage_source.html @@ -226,7 +226,8 @@
  • Grid Converters
    • diff --git a/docs/_modules/motulator/grid/utils/_plots.html b/docs/_modules/motulator/grid/utils/_plots.html index 09613ab3..ff6d8721 100644 --- a/docs/_modules/motulator/grid/utils/_plots.html +++ b/docs/_modules/motulator/grid/utils/_plots.html @@ -226,7 +226,8 @@
  • Grid Converters
    • @@ -559,12 +560,13 @@

    Source code for motulator.grid.utils._plots

             "--",
         label=r"$p_\mathrm{g,ref}$",
         ds="steps-post")
-    ax1.plot(
-        ctrl.t,
-        ctrl.ref.q_g/base.p,
-        "--",
-        label=r"$q_\mathrm{g,ref}$",
-        ds="steps-post")
+    if hasattr(ctrl.ref, "q_g"):
+        ax1.plot(
+            ctrl.t,
+            ctrl.ref.q_g/base.p,
+            "--",
+            label=r"$q_\mathrm{g,ref}$",
+            ds="steps-post")
     ax1.legend()
     ax1.set_xlim(t_span)
     ax1.set_xticklabels([])
diff --git a/docs/_modules/motulator/grid/utils/_utils.html b/docs/_modules/motulator/grid/utils/_utils.html
index f3d50cb1..dd56304c 100644
--- a/docs/_modules/motulator/grid/utils/_utils.html
+++ b/docs/_modules/motulator/grid/utils/_utils.html
@@ -226,7 +226,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/_sources/autoapi/motulator/grid/control/index.rst b/docs/_sources/autoapi/motulator/grid/control/index.rst
index a14b3fd8..01ece6d5 100644
--- a/docs/_sources/autoapi/motulator/grid/control/index.rst
+++ b/docs/_sources/autoapi/motulator/grid/control/index.rst
@@ -315,8 +315,6 @@ Package Contents
                     u_cs : complex
                         Realized converter output voltage (V) in stationary
                         coordinates. This signal is obtained from the PWM.
-                    u_gs : complex
-                        PCC voltage (V) in stationary coordinates.
       :rtype: SimpleNamespace
 
 
@@ -373,7 +371,7 @@ Package Contents
 
       :returns: **ref** --
 
-                Reference signals, containing the following fields:
+                Reference signals, containing some of the following fields:
 
                     u_dc : float
                         DC-bus voltage reference (V).
@@ -591,7 +589,7 @@ Package Contents
    This implements the RFPSC-type grid-forming mode of the control method
    described in [#Nur2024]_. Transparent current control is also implemented.
 
-   :param cfg: Controller configuration parameters.
+   :param cfg: Control system configuration parameters.
    :type cfg: ObserverBasedGridFormingControlCfg
 
    .. rubric:: Notes
@@ -713,12 +711,14 @@ Package Contents
    :type R: float, optional
    :param R_a: Active resistance (Ω). The default is 0.25*nom_u/max_i.
    :type R_a: float, optional
-   :param T_s: Sampling period (s). The default is 100e-6.
-   :type T_s: float, optional
+   :param k_v: Voltage control gain. The default is `alpha_o/nom_w`.
+   :type k_v: float, optional
    :param alpha_c: Current control bandwidth (rad/s). The default is 2*pi*400.
    :type alpha_c: float, optional
    :param alpha_o: Observer gain (rad/s). The default is 2*pi*50.
    :type alpha_o: float, optional
+   :param T_s: Sampling period (s). The default is 100e-6.
+   :type T_s: float, optional
 
 
 
@@ -834,8 +834,8 @@ Package Contents
    .. rubric:: References
 
    .. [#Har2020] Harnefors, Rahman, Hinkkanen, Routimo, "Reference-feedforward
-       power-synchronization control," IEEE Trans. Power Electron., 2020,
-       https://doi.org/10.1109/TPEL.2020.2970991
+      power-synchronization control," IEEE Trans. Power Electron., 2020,
+      https://doi.org/10.1109/TPEL.2020.2970991
 
 
 
@@ -937,12 +937,14 @@ Package Contents
    :type nom_w: float
    :param max_i: Maximum current (A), peak value.
    :type max_i: float
+   :param R: Total series resistance (Ω). The default is 0.
+   :type R: float, optional
    :param R_a: Active resistance (Ω). The default is 0.25*nom_u/max_i.
    :type R_a: float, optional
-   :param T_s: Sampling period (s). The default is 100e-6.
-   :type T_s: float, optional
    :param w_b: Low-pass filter bandwidth (rad/s). The default is 2*pi*5.
    :type w_b: float, optional
+   :param T_s: Sampling period (s). The default is 100e-6.
+   :type T_s: float, optional
 
 
 
diff --git a/docs/_sources/control/control_system.rst b/docs/_sources/control/control_system.rst
index 4b3c4c3b..a725ccbb 100644
--- a/docs/_sources/control/control_system.rst
+++ b/docs/_sources/control/control_system.rst
@@ -4,11 +4,11 @@ Common
 Data Flow
 ---------
 
-The two figures below show the structure and data flow in a typical simulation model as well as an example of the internal structure of a discrete-time control system. The text in italics refers to the default object names used in the software. In discrete-time control systems, the signals are collected into the following :class:`types.SimpleNamespace` objects: 
+The two figures below show the structure and data flow in a typical simulation model as well as an example of the internal structure of a discrete-time control system. The text in italics refers to the default object names used in the software. In discrete-time control systems, the signals are collected into the following :class:`types.SimpleNamespace` objects:
 
-- The object `fbk` contains feedback signals for the controllers. These signals can be measured quantities (such as the measured DC-bus voltage `fbk.u_dc`) as well as estimated quantities (such as the estimated stator flux linkage `fbk.psi_s`). 
+- The object `fbk` contains feedback signals for the controllers. These signals can be measured quantities (such as the measured DC-bus voltage `fbk.u_dc`) as well as estimated quantities (such as the estimated stator flux linkage `fbk.psi_s`).
 
-- The object `ref` contains the reference signals, both externally provided (such as `ref.w_m` for the angular speed reference) and internally generated (such as `ref.d_c_abc` for the duty ratios). 
+- The object `ref` contains the reference signals, both externally provided (such as `ref.w_m` for the angular speed reference) and internally generated (such as `ref.d_c_abc` for the duty ratios).
 
 These objects `fbk` and `ref` may propagate through several blocks (implemented as classes in the software), which may add new signals or modify the existing one. At the end of their propagation chain, both objects `fbk` and `ref` are saved to the lists. Therefore, the most relevant internal signals of the control system are directly accessible after the simulation. Furthermore, the states and inputs of the continuous-time system model are also saved. In the post-processing stage, the saved data is converted to the NumPy arrays.
 
@@ -16,22 +16,22 @@ These objects `fbk` and `ref` may propagate through several blocks (implemented
    :width: 100%
    :align: center
    :alt: Block diagram of the control system.
-   
-   Block diagram illustrating the structure and data flow in a typical simulation model. The lower part of the figure illustrates how the data is saved. The post-processing is done after the simulation. The internal structure of a typical control system is exemplified in the figure below. 
+
+   Block diagram illustrating the structure and data flow in a typical simulation model. The lower part of the figure illustrates how the data is saved. The post-processing is done after the simulation. The internal structure of a typical control system is exemplified in the figure below.
 
 .. figure:: figs/discrete_control_system.svg
    :width: 100%
    :align: center
    :alt: Block diagram of the control system.
-   
-   Block diagram exemplifying the internal structure of a typical cascade control system. The object `ref` at the control system output should contain the sampling period `T_s` and the converter duty ratios `d_c_abc` for the carrier comparison. The observer does not necessarily exist in all control systems (or it can be replaced with, e.g., a phase-locked loop). 
+
+   Block diagram exemplifying the internal structure of a typical cascade control system. The object `ref` at the control system output should contain the sampling period `T_s` and the converter duty ratios `d_c_abc` for the carrier comparison. The observer does not necessarily exist in all control systems (or it can be replaced with, e.g., a phase-locked loop).
 
 Main Control Loop
 -----------------
 
 A template for the main control loop is available in the base class for discrete-time control systems in :class:`motulator.common.control.ControlSystem`. The main control loop in this template has the following steps:
 
-1. Get the feedback signals `fbk` for the controllers from the outputs of the continuous-time system model `mdl`. This step may contain first getting the measurements and then optionally computing the observer outputs (or otherwise manipulating the measured signals).  
+1. Get the feedback signals `fbk` for the controllers from the outputs of the continuous-time system model `mdl`. This step may contain first getting the measurements and then optionally computing the observer outputs (or otherwise manipulating the measured signals).
 2. Get the reference signals `ref` and compute the controller outputs based on the feedback signals `fbk`. Cascade control systems may contain multiple controllers, where the output of the outer controller is the reference signal for the inner controller.
 3. Update the states of the control system for the next sampling instant.
 4. Save the feedback signals `fbk` and the reference signals `ref` so they can be accessed after the simulation.
@@ -54,14 +54,14 @@ The figure below shows a 2DOF PI controller with an optional feedforward term. I
    u &= k_\mathrm{t}r - k_\mathrm{p}y + u_\mathrm{i} + u_\mathrm{ff}
    :label: 2dof_pi
 
-where :math:`r` is the reference signal, :math:`y` is the measured (or estimated) feedback signal, :math:`u_\mathrm{i}` is the the integral state, and :math:`u_\mathrm{ff}` is the optional feedforward signal. Furthermore, :math:`k_\mathrm{t}` is the reference-feedforward gain, :math:`k_\mathrm{p}` is the proportional gain, and :math:`k_\mathrm{i}` is the integral gain. Setting :math:`k_\mathrm{t} = k_\mathrm{p}` and :math:`u_\mathrm{ff} = 0` results in the standard PI controller. This 2DOF PI controller can also be understood as a state-feedback controller with integral action and reference feedforward [#Fra1997]_. 
+where :math:`r` is the reference signal, :math:`y` is the measured (or estimated) feedback signal, :math:`u_\mathrm{i}` is the the integral state, and :math:`u_\mathrm{ff}` is the optional feedforward signal. Furthermore, :math:`k_\mathrm{t}` is the reference-feedforward gain, :math:`k_\mathrm{p}` is the proportional gain, and :math:`k_\mathrm{i}` is the integral gain. Setting :math:`k_\mathrm{t} = k_\mathrm{p}` and :math:`u_\mathrm{ff} = 0` results in the standard PI controller. This 2DOF PI controller can also be understood as a state-feedback controller with integral action and reference feedforward [#Fra1997]_.
 
 .. figure:: figs/2dof_pi.svg
    :width: 100%
    :align: center
    :alt: 2DOF PI controller.
 
-   2DOF PI controller with an optional feedforward term. The operator :math:`1/s` refers to integration. A discrete-time variant of this controller with the integrator anti-windup is implemented in the :class:`motulator.common.control.PIController` class.  
+   2DOF PI controller with an optional feedforward term. The operator :math:`1/s` refers to integration. A discrete-time variant of this controller with the integrator anti-windup is implemented in the :class:`motulator.common.control.PIController` class.
 
 Disturbance-Observer Structure
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
@@ -71,7 +71,7 @@ The controller :eq:`2dof_pi` can be equally represented using the disturbance-ob
 .. math::
 	\frac{\mathrm{d} u_\mathrm{i}}{\mathrm{d} t} &= \alpha_\mathrm{i}\left(u - \hat v\right) \\
    \hat v &= u_\mathrm{i} - (k_\mathrm{p} - k_\mathrm{t})y + u_\mathrm{ff} \\
-   u &= k_\mathrm{t}\left(r - y\right) + \hat v 
+   u &= k_\mathrm{t}\left(r - y\right) + \hat v
    :label: 2dof_pi_disturbance_observer
 
 where :math:`\alpha_\mathrm{i} = k_\mathrm{i}/k_\mathrm{t}` is the redefined integral gain and :math:`\hat v` is the input-equivalent disturbance estimate. This structure is convenient to prevent the integral windup that originates from the actuator saturation [#Fra1997]_. The actuator output is limited in practice due to physical constraints. Consequently, the realized actuator output is
@@ -85,7 +85,7 @@ where :math:`\mathrm{sat}(\cdot)` is the saturation function. If this saturation
 	\frac{\mathrm{d} u_\mathrm{i}}{\mathrm{d} t} = \alpha_\mathrm{i}\left(\bar{u} - \hat v \right)
    :label: anti_windup
 
-The other parts of the above controller are not affected by the saturation. 
+The other parts of the above controller are not affected by the saturation.
 
 Discrete-Time Algorithm
 ^^^^^^^^^^^^^^^^^^^^^^^
@@ -99,7 +99,7 @@ The discrete-time variant of the controller :eq:`2dof_pi_disturbance_observer` w
    \bar{u}(k) &= \mathrm{sat}[u(k)]
    :label: discrete_2dof_pi
 
-where :math:`T_\mathrm{s}` is the sampling period and :math:`k` is the discrete-time index. This algorithm corresponds to the actual implementation in the :class:`motulator.common.control.PIController` class. 
+where :math:`T_\mathrm{s}` is the sampling period and :math:`k` is the discrete-time index. This algorithm corresponds to the actual implementation in the :class:`motulator.common.control.PIController` class.
 
 .. _complex-vector-2dof-pi-controller:
 
@@ -113,7 +113,7 @@ As shown in the figure below, the 2DOF PI controller presented above can be exte
     \boldsymbol{u} &= \boldsymbol{k}_\mathrm{t}\boldsymbol{r} - \boldsymbol{k}_\mathrm{p}\boldsymbol{y} + \boldsymbol{u}_\mathrm{i} + \boldsymbol{u}_\mathrm{ff}
     :label: complex_vector_2dof_pi
 
-where :math:`\boldsymbol{u}` is the output of the controller, :math:`\boldsymbol{r}` is the reference signal, :math:`\boldsymbol{u}_\mathrm{i}` is the the integral state, and :math:`\boldsymbol{u}_\mathrm{ff}` is the optional feedforward signal. Furthermore, :math:`\boldsymbol{k}_\mathrm{t}` is the reference-feedforward gain, :math:`\boldsymbol{k}_\mathrm{p}` is the proportional gain, and :math:`\boldsymbol{k}_\mathrm{i}` is the integral gain. 
+where :math:`\boldsymbol{u}` is the output of the controller, :math:`\boldsymbol{r}` is the reference signal, :math:`\boldsymbol{u}_\mathrm{i}` is the the integral state, and :math:`\boldsymbol{u}_\mathrm{ff}` is the optional feedforward signal. Furthermore, :math:`\boldsymbol{k}_\mathrm{t}` is the reference-feedforward gain, :math:`\boldsymbol{k}_\mathrm{p}` is the proportional gain, and :math:`\boldsymbol{k}_\mathrm{i}` is the integral gain.
 
 .. figure:: figs/complex_vector_2dof_pi.svg
    :width: 100%
@@ -131,5 +131,3 @@ The discrete-time implementation of :eq:`complex_vector_2dof_pi` with the anti-w
 .. [#Fra1997] Franklin, Powell, Workman, "Digital Control of Dynamic Systems," 3rd ed., Menlo Park, CA: Addison-Wesley, 1997
 
 .. [#Bri1999] Briz del Blanco, Degner, Lorenz, “Dynamic analysis of current regulators for AC motors using complex vectors,” IEEE Trans. Ind. Appl., 1999, https://doi.org/10.1109/28.806058
-
-
diff --git a/docs/_sources/control/drive/current_ctrl.rst b/docs/_sources/control/drive/current_ctrl.rst
index 4a6dd676..98ba18a1 100644
--- a/docs/_sources/control/drive/current_ctrl.rst
+++ b/docs/_sources/control/drive/current_ctrl.rst
@@ -1,7 +1,7 @@
 Current Control
 ===============
 
-Synchronous-frame two-degrees-of-freedom (2DOF) proportional-integral (PI) current control is commonly used in three-phase AC machine drives [#Har1998]_, [#Bri1999]_, [#Awa2019]_, [#Hin2024]_. This control structure allows to compensate for the cross-coupling originating from rotating coordinates as well as to improve disturbance rejection. 
+Synchronous-frame two-degrees-of-freedom (2DOF) proportional-integral (PI) current control is commonly used in three-phase AC machine drives [#Har1998]_, [#Bri1999]_, [#Awa2019]_, [#Hin2024]_. This control structure allows to compensate for the cross-coupling originating from rotating coordinates as well as to improve disturbance rejection.
 
 A 2DOF PI current controller for induction machines is available in the :class:`motulator.drive.control.im.CurrentController` class and for synchronous machines in the :class:`motulator.drive.control.sm.CurrentController` class, both of which inherit from the :class:`motulator.common.control.ComplexPIController` class. In the following, current control of induction machines is first considered in detail. Then, the same principles are applied to synchronous machines. Complex space vectors are used to represent three-phase quantities.
 
@@ -15,7 +15,7 @@ The inverse-Γ model of an induction machine is considered (see :doc:`/model/dri
 
 .. math::
     L_\sigma \frac{\mathrm{d} \boldsymbol{i}_\mathrm{s}}{\mathrm{d} t} &= \boldsymbol{u}_\mathrm{s} - (R_\sigma + \mathrm{j} \omega_\mathrm{s}L_\sigma)\boldsymbol{i}_\mathrm{s} - \underbrace{\left(\mathrm{j}\omega_\mathrm{m} - \frac{R_\mathrm{R}}{L_\mathrm{M}}\right)\boldsymbol{\psi}_\mathrm{R}}_{\text{back-emf } \boldsymbol{e}_\mathrm{s}} \\
-	\frac{\mathrm{d} \boldsymbol{\psi}_\mathrm{R}}{\mathrm{d} t} &= R_\mathrm{R}\boldsymbol{i}_\mathrm{s} - \left(\frac{R_\mathrm{R}}{L_\mathrm{M}} + \mathrm{j}\omega_\mathrm{r} \right)\boldsymbol{\psi}_\mathrm{R} 
+	\frac{\mathrm{d} \boldsymbol{\psi}_\mathrm{R}}{\mathrm{d} t} &= R_\mathrm{R}\boldsymbol{i}_\mathrm{s} - \left(\frac{R_\mathrm{R}}{L_\mathrm{M}} + \mathrm{j}\omega_\mathrm{r} \right)\boldsymbol{\psi}_\mathrm{R}
 
 where :math:`R_\sigma = R_\mathrm{s} + R_\mathrm{R}` is the total resistance and :math:`\omega_\mathrm{r} = \omega_\mathrm{s} - \omega_\mathrm{m}` is the slip angular frequency. The rotor flux linkage :math:`\boldsymbol{\psi}_\mathrm{R}` and the rotor speed :math:`\omega_\mathrm{m}` change slowly as compared to the stator current. Consequently, the back-emf :math:`\boldsymbol{e}_\mathrm{s}` can be considered as a quasi-constant input (load) disturbance for the current controller, and it suffices to consider the stator current dynamics
 
@@ -30,15 +30,15 @@ The design of synchronous-frame 2DOF PI current control is considered in the con
 
 .. math::
 	\frac{\mathrm{d} \boldsymbol{u}_\mathrm{i}}{\mathrm{d} t} &= (\boldsymbol{k}_\mathrm{i} + \mathrm{j}\omega_\mathrm{s}\boldsymbol{k}_\mathrm{t} )\left(\boldsymbol{i}_\mathrm{s,ref} - \boldsymbol{i}_\mathrm{s}\right) \\
-    \boldsymbol{u}_\mathrm{s,ref} &= \boldsymbol{k}_\mathrm{t}\boldsymbol{i}_\mathrm{s,ref} - \boldsymbol{k}_\mathrm{p}\boldsymbol{i}_\mathrm{s} + \boldsymbol{u}_\mathrm{i} 
+    \boldsymbol{u}_\mathrm{s,ref} &= \boldsymbol{k}_\mathrm{t}\boldsymbol{i}_\mathrm{s,ref} - \boldsymbol{k}_\mathrm{p}\boldsymbol{i}_\mathrm{s} + \boldsymbol{u}_\mathrm{i}
     :label: cc
 
-where :math:`\boldsymbol{u}_\mathrm{s,ref}` is the output of the controller, i.e., the stator voltage reference, :math:`\boldsymbol{i}_\mathrm{s,ref}` is the stator current reference, :math:`\boldsymbol{u}_\mathrm{i}` is the the integral state, and :math:`\omega_\mathrm{s}` is the angular speed of the coordinate system. Furthermore, :math:`\boldsymbol{k}_\mathrm{t}` is the reference-feedforward gain, :math:`\boldsymbol{k}_\mathrm{p}` is the proportional gain, and :math:`\boldsymbol{k}_\mathrm{i}` is the integral gain. 
+where :math:`\boldsymbol{u}_\mathrm{s,ref}` is the output of the controller, i.e., the stator voltage reference, :math:`\boldsymbol{i}_\mathrm{s,ref}` is the stator current reference, :math:`\boldsymbol{u}_\mathrm{i}` is the the integral state, and :math:`\omega_\mathrm{s}` is the angular speed of the coordinate system. Furthermore, :math:`\boldsymbol{k}_\mathrm{t}` is the reference-feedforward gain, :math:`\boldsymbol{k}_\mathrm{p}` is the proportional gain, and :math:`\boldsymbol{k}_\mathrm{i}` is the integral gain.
 
 .. note::
-   The gain definitions used in :eq:`cc` differ from [#Hin2024]_, where a more general controller structure is considered. 
+   The gain definitions used in :eq:`cc` differ from [#Hin2024]_, where a more general controller structure is considered.
 
-Closed-Loop System 
+Closed-Loop System
 ^^^^^^^^^^^^^^^^^^
 
 Here, ideal voltage production is assumed, :math:`\boldsymbol{u}_\mathrm{s} = \boldsymbol{u}_\mathrm{s,ref}`. Using :eq:`im_current` and :eq:`cc`, the closed-loop system in the Laplace domain becomes
@@ -49,13 +49,13 @@ Here, ideal voltage production is assumed, :math:`\boldsymbol{u}_\mathrm{s} = \b
 The disturbance rejection depends on the closed-loop admittance
 
 .. math::
-    \boldsymbol{Y}_\mathrm{c}(s) = \frac{s}{L_\sigma s^2 + (R_\sigma + \mathrm{j}\omega_\mathrm{s} L_\sigma + \boldsymbol{k}_\mathrm{p}) s + \boldsymbol{k}_\mathrm{i} + \mathrm{j}\omega_\mathrm{s} \boldsymbol{k}_\mathrm{t}} 
+    \boldsymbol{Y}_\mathrm{c}(s) = \frac{s}{L_\sigma s^2 + (R_\sigma + \mathrm{j}\omega_\mathrm{s} L_\sigma + \boldsymbol{k}_\mathrm{p}) s + \boldsymbol{k}_\mathrm{i} + \mathrm{j}\omega_\mathrm{s} \boldsymbol{k}_\mathrm{t}}
     :label: Yc
 
 The closed-loop poles can be arbitrarily placed by means of the gains. The reference-tracking transfer function is
 
 .. math::
-	\boldsymbol{G}_\mathrm{c}(s) = \frac{(s + \mathrm{j}\omega_\mathrm{s}) \boldsymbol{k}_\mathrm{t} + \boldsymbol{k}_\mathrm{i} }{L_\sigma s^2 + (R_\sigma + \mathrm{j}\omega_\mathrm{s} L_\sigma + \boldsymbol{k}_\mathrm{p}) s + \boldsymbol{k}_\mathrm{i} + \mathrm{j}\omega_\mathrm{s} \boldsymbol{k}_\mathrm{t}}     
+	\boldsymbol{G}_\mathrm{c}(s) = \frac{(s + \mathrm{j}\omega_\mathrm{s}) \boldsymbol{k}_\mathrm{t} + \boldsymbol{k}_\mathrm{i} }{L_\sigma s^2 + (R_\sigma + \mathrm{j}\omega_\mathrm{s} L_\sigma + \boldsymbol{k}_\mathrm{p}) s + \boldsymbol{k}_\mathrm{i} + \mathrm{j}\omega_\mathrm{s} \boldsymbol{k}_\mathrm{t}}
     :label: Gc
 
 whose zero can be placed by means of the reference-feedforward gain :math:`\boldsymbol{k}_\mathrm{t}`.
@@ -65,7 +65,7 @@ Gain Selection
 
 Consider the gains
 
-.. math::                
+.. math::
     \boldsymbol{k}_\mathrm{p} = 2\alpha_\mathrm{c} \hat L_\sigma - \hat R_\sigma \qquad\qquad
     \boldsymbol{k}_\mathrm{i} = \alpha_\mathrm{c}^2\hat L_\sigma  \qquad \qquad
     \boldsymbol{k}_\mathrm{t} = \alpha_\mathrm{c} \hat L_\sigma
@@ -76,9 +76,9 @@ where :math:`\hat R_\sigma = 0` can be used in practice due to its minor effect
 .. math::
     \boldsymbol{Y}_\mathrm{c}(s) = \frac{s}{L_\sigma (s + \alpha_\mathrm{c})(s + \alpha_\mathrm{c} + \mathrm{j}\omega_\mathrm{s})}
     \qquad\qquad
-    \boldsymbol{G}_\mathrm{c}(s) = \frac{\alpha_\mathrm{c}}{s + \alpha_\mathrm{c}} 
+    \boldsymbol{G}_\mathrm{c}(s) = \frac{\alpha_\mathrm{c}}{s + \alpha_\mathrm{c}}
 
-It can be seen that this design results in the first-order reference-tracking dynamics. Furthermore, one pole is placed at the real axis at :math:`s=-\alpha_\mathrm{c}`, while another pole moves with the angular frequency of the coordinate system, :math:`s= -\alpha_\mathrm{c} - \mathrm{j}\omega_\mathrm{s}`. The complex-vector design tends to be slightly more robust to parameter errors than the IMC design since the other closed-loop pole approximately corresponds to the open-loop pole.  
+It can be seen that this design results in the first-order reference-tracking dynamics. Furthermore, one pole is placed at the real axis at :math:`s=-\alpha_\mathrm{c}`, while another pole moves with the angular frequency of the coordinate system, :math:`s= -\alpha_\mathrm{c} - \mathrm{j}\omega_\mathrm{s}`. The complex-vector design tends to be slightly more robust to parameter errors than the IMC design since the other closed-loop pole approximately corresponds to the open-loop pole.
 
 This gain selection is used in the :class:`motulator.drive.control.im.CurrentController` class. The stator voltage is limited in practice due to the limited DC-bus voltage of the converter. Consequently, the realized (limited) voltage reference is
 
@@ -93,13 +93,13 @@ Synchronous Machines
 System Model
 ^^^^^^^^^^^^
 
-Consider the synchronous machine model in rotor coordinates, rotating at :math:`\omega_\mathrm{m}` and aligned along the d-axis of the rotor, 
+Consider the synchronous machine model in rotor coordinates, rotating at :math:`\omega_\mathrm{m}` and aligned along the d-axis of the rotor,
 
 .. math::
     \frac{\mathrm{d}\boldsymbol{\psi}_\mathrm{s}}{\mathrm{d} t} &= \boldsymbol{u}_\mathrm{s} - R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - \mathrm{j}\omega_\mathrm{m}\boldsymbol{\psi}_\mathrm{s} \\
- 	\boldsymbol{\psi}_\mathrm{s} &= L_\mathrm{d}\mathrm{Re}\{\boldsymbol{i}_\mathrm{s}\} + \mathrm{j}L_\mathrm{q}\mathrm{Im}\{\boldsymbol{i}_\mathrm{s}\} + \psi_\mathrm{f} 
+ 	\boldsymbol{\psi}_\mathrm{s} &= L_\mathrm{d}\mathrm{Re}\{\boldsymbol{i}_\mathrm{s}\} + \mathrm{j}L_\mathrm{q}\mathrm{Im}\{\boldsymbol{i}_\mathrm{s}\} + \psi_\mathrm{f}
     :label: sm_model
- 
+
 where linear magnetics are assumed for simplicity. For further details, see :doc:`/model/drive/machines`.
 
 2DOF PI Controller
@@ -109,28 +109,28 @@ An internal change of the state variable from the stator current to the stator f
 
 .. math::
     \boldsymbol{\psi}_\mathrm{s,ref} &= \hat{L}_\mathrm{d}\mathrm{Re}\{\boldsymbol{i}_\mathrm{s,ref}\} + \mathrm{j} \hat{L}_\mathrm{q}\mathrm{Im}\{\boldsymbol{i}_\mathrm{s,ref}\} \\
-    \hat{\boldsymbol{\psi}_\mathrm{s}} &= \hat{L}_\mathrm{d}\mathrm{Re}\{\boldsymbol{i}_\mathrm{s}\} + \mathrm{j} \hat{L}_\mathrm{q}\mathrm{Im}\{\boldsymbol{i}_\mathrm{s}\} 
+    \hat{\boldsymbol{\psi}_\mathrm{s}} &= \hat{L}_\mathrm{d}\mathrm{Re}\{\boldsymbol{i}_\mathrm{s}\} + \mathrm{j} \hat{L}_\mathrm{q}\mathrm{Im}\{\boldsymbol{i}_\mathrm{s}\}
     :label: flux_mapping_sm
 
-This choice of using the flux linkage as the internal state has some advantages: the gain expressions become simpler; the magnetic saturation would be more convenient to take into account; and the same control structure can be used for salient and nonsalient machines. 
+This choice of using the flux linkage as the internal state has some advantages: the gain expressions become simpler; the magnetic saturation would be more convenient to take into account; and the same control structure can be used for salient and nonsalient machines.
 
-Here, the complex vector design is considered. Hence, the controller :eq:`cc` can be rewritten as 
+Here, the complex vector design is considered. Hence, the controller :eq:`cc` can be rewritten as
 
 .. math::
 	\frac{\mathrm{d} \boldsymbol{u}_\mathrm{i}}{\mathrm{d} t} &= (\boldsymbol{k}_\mathrm{i} + \mathrm{j}\omega_\mathrm{s}\boldsymbol{k}_\mathrm{t} )\left(\boldsymbol{\psi}_\mathrm{s,ref} - \hat{\boldsymbol{\psi}}_\mathrm{s}\right) \\
-    \boldsymbol{u}_\mathrm{s,ref} &= \boldsymbol{k}_\mathrm{t}\boldsymbol{\psi}_\mathrm{s,ref} - \boldsymbol{k}_\mathrm{p}\hat{\boldsymbol{\psi}}_\mathrm{s} + \boldsymbol{u}_\mathrm{i} 
+    \boldsymbol{u}_\mathrm{s,ref} &= \boldsymbol{k}_\mathrm{t}\boldsymbol{\psi}_\mathrm{s,ref} - \boldsymbol{k}_\mathrm{p}\hat{\boldsymbol{\psi}}_\mathrm{s} + \boldsymbol{u}_\mathrm{i}
     :label: cc_flux
 
-where the angular speed of the coordinate system equals typically the measured rotor speed, :math:`\omega_\mathrm{s} = \omega_\mathrm{m}`, or the estimated rotor speed :math:`\omega_\mathrm{s} = \hat{\omega}_\mathrm{m}`. If the magnetic saturation is not considered, this flux-linkage-based current controller is equivalent to a regular 2DOF PI current controller (even if inductance estimates are inaccurate). Notice that :math:`\boldsymbol{i}_\mathrm{s,ref} = \boldsymbol{i}_\mathrm{s}` holds in the steady state even with inductance estimate inaccuracies, since the same inductances are used to map both the reference current and the actual current to the corresponding flux linkages. 
+where the angular speed of the coordinate system equals typically the measured rotor speed, :math:`\omega_\mathrm{s} = \omega_\mathrm{m}`, or the estimated rotor speed :math:`\omega_\mathrm{s} = \hat{\omega}_\mathrm{m}`. If the magnetic saturation is not considered, this flux-linkage-based current controller is equivalent to a regular 2DOF PI current controller (even if inductance estimates are inaccurate). Notice that :math:`\boldsymbol{i}_\mathrm{s,ref} = \boldsymbol{i}_\mathrm{s}` holds in the steady state even with inductance estimate inaccuracies, since the same inductances are used to map both the reference current and the actual current to the corresponding flux linkages.
 
 The gain selection analogous to :eq:`complex_vector_gains` becomes
 
-.. math::                
+.. math::
     \boldsymbol{k}_\mathrm{p} = 2\alpha_\mathrm{c} - \hat R_\mathrm{s} \qquad\qquad
     \boldsymbol{k}_\mathrm{i} = \alpha_\mathrm{c}^2 \qquad \qquad
-    \boldsymbol{k}_\mathrm{t} = \alpha_\mathrm{c} 
+    \boldsymbol{k}_\mathrm{t} = \alpha_\mathrm{c}
 
-where :math:`\hat R_\mathrm{s} = 0` can be used in practice. Assume accurate parameter estimates and perfect alignment of the controller coordinate system with the rotor coordinate system. Then, using :eq:`sm_model`, :eq:`flux_mapping_sm`, and :eq:`cc_flux`, the closed-loop system can be shown to be analogous to the induction machine case. This control design corresponds to the implementation in the :class:`motulator.drive.control.sm.CurrentController` class. 
+where :math:`\hat R_\mathrm{s} = 0` can be used in practice. Assume accurate parameter estimates and perfect alignment of the controller coordinate system with the rotor coordinate system. Then, using :eq:`sm_model`, :eq:`flux_mapping_sm`, and :eq:`cc_flux`, the closed-loop system can be shown to be analogous to the induction machine case. This control design corresponds to the implementation in the :class:`motulator.drive.control.sm.CurrentController` class.
 
 .. rubric:: References
 
@@ -141,5 +141,3 @@ where :math:`\hat R_\mathrm{s} = 0` can be used in practice. Assume accurate par
 .. [#Awa2019] Awan, Saarakkala, Hinkkanen, "Flux-linkage-based current control of saturated synchronous motors," IEEE Trans. Ind. Appl. 2019, https://doi.org/10.1109/TIA.2019.2919258
 
 .. [#Hin2024] Hinkkanen,  Harnefors, Kukkola, "Fundamentals of Electric Machine Drives," lecture notes, 2024, https://doi.org/10.5281/zenodo.10609166
-
-
diff --git a/docs/_sources/control/drive/observers.rst b/docs/_sources/control/drive/observers.rst
index 59845c23..095084ab 100644
--- a/docs/_sources/control/drive/observers.rst
+++ b/docs/_sources/control/drive/observers.rst
@@ -15,10 +15,10 @@ The inverse-Γ model of an induction machine is considered (see :doc:`/model/dri
 
 .. math::
 	\frac{\mathrm{d} \boldsymbol{\psi}_\mathrm{R}}{\mathrm{d} t} + \mathrm{j}\omega_\mathrm{s}\boldsymbol{\psi}_\mathrm{R} &= \boldsymbol{u}_\mathrm{s} - R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - L_\sigma \frac{\mathrm{d} \boldsymbol{i}_\mathrm{s}}{\mathrm{d} t} - \mathrm{j} \omega_\mathrm{s}L_\sigma\boldsymbol{i}_\mathrm{s} \\
-    &= R_\mathrm{R}\boldsymbol{i}_\mathrm{s} - \left(\alpha - \mathrm{j}\omega_\mathrm{m} \right)\boldsymbol{\psi}_\mathrm{R}
+    &= \underbrace{R_\mathrm{R}\boldsymbol{i}_\mathrm{s} - \left(\alpha - \mathrm{j}\omega_\mathrm{m} \right)\boldsymbol{\psi}_\mathrm{R}}_{\boldsymbol{v}}
     :label: dpsiR
 
-where :math:`\omega_\mathrm{m}` is the electrical rotor angular speed.
+where :math:`\omega_\mathrm{m}` is the electrical rotor angular speed and :math:`\alpha = R_\mathrm{R}/L_\mathrm{M}` is the inverse rotor time constant.
 
 Observer Structure
 ^^^^^^^^^^^^^^^^^^
@@ -29,20 +29,20 @@ General Coordinates
 Based on :eq:`dpsiR`, a reduced-order observer can be formulated in a coordinate system rotating at :math:`\omega_\mathrm{s}`,
 
 .. math::
-    \frac{\mathrm{d} \hat{\boldsymbol{\psi}}_\mathrm{R}}{\mathrm{d} t} + \mathrm{j}\omega_\mathrm{s}\hat{\boldsymbol{\psi}}_\mathrm{R} = \boldsymbol{v} + \boldsymbol{k}_1(\hat{\boldsymbol{v}} - \boldsymbol{v}) + \boldsymbol{k}_2(\hat{\boldsymbol{v}} - \boldsymbol{v})^*
+    \frac{\mathrm{d} \hat{\boldsymbol{\psi}}_\mathrm{R}}{\mathrm{d} t} + \mathrm{j}\omega_\mathrm{s}\hat{\boldsymbol{\psi}}_\mathrm{R} = \hat{\boldsymbol{v}}_\mathrm{s} + \boldsymbol{k}_1(\hat{\boldsymbol{v}} - \hat{\boldsymbol{v}}_\mathrm{s}) + \boldsymbol{k}_2(\hat{\boldsymbol{v}} - \hat{\boldsymbol{v}}_\mathrm{s})^*
     :label: dhatpsiR
 
 where :math:`\boldsymbol{k}_1` and :math:`\boldsymbol{k}_2` are complex gains, the estimates are marked with the hat, and :math:`^*` marks the complex conjugate. The back-emf estimates are
 
 .. math::
-    \boldsymbol{v} &= \boldsymbol{u}_\mathrm{s} - R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - L_\sigma \frac{\mathrm{d} \boldsymbol{i}_\mathrm{s}}{\mathrm{d} t} - \mathrm{j} \omega_\mathrm{s}L_\sigma\boldsymbol{i}_\mathrm{s} \\
-	\hat{\boldsymbol{v}} &= R_\mathrm{R}\boldsymbol{i}_\mathrm{s} - \left(\alpha - \mathrm{j}\hat{\omega}_\mathrm{m} \right)\hat{\boldsymbol{\psi}}_\mathrm{R}
+    \hat{\boldsymbol{v}}_\mathrm{s} &= \boldsymbol{u}_\mathrm{s} - \hat R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - \hat L_\sigma \frac{\mathrm{d} \boldsymbol{i}_\mathrm{s}}{\mathrm{d} t} - \mathrm{j} \omega_\mathrm{s}\hat L_\sigma\boldsymbol{i}_\mathrm{s} \\
+	\hat{\boldsymbol{v}} &= \hat R_\mathrm{R}\boldsymbol{i}_\mathrm{s} - \left(\hat \alpha - \mathrm{j}\hat{\omega}_\mathrm{m} \right)\hat{\boldsymbol{\psi}}_\mathrm{R}
     :label: hatv
 
-In sensored drives, the rotor speed estimate is replaced with the measured speed,  :math:`\hat{\omega}_\mathrm{m} = \omega_\mathrm{m}`. If needed, the estimates for the stator flux linkage and the electromagnetic torque, respectively, are
+In sensored drives, the rotor speed estimate is replaced with the measured speed, :math:`\hat{\omega}_\mathrm{m} = \omega_\mathrm{m}`. If needed, the estimates for the stator flux linkage and the electromagnetic torque, respectively, are
 
 .. math::
-    \hat{\boldsymbol{\psi}}_\mathrm{s} &= \hat{\boldsymbol{\psi}}_\mathrm{R} + L_\sigma \boldsymbol{i}_\mathrm{s} \\
+    \hat{\boldsymbol{\psi}}_\mathrm{s} &= \hat{\boldsymbol{\psi}}_\mathrm{R} + \hat L_\sigma \boldsymbol{i}_\mathrm{s} \\
     \hat{\tau}_\mathrm{M} &= \frac{3n_\mathrm{p}}{2}\mathrm{Im}\left\{\boldsymbol{i}_\mathrm{s} \hat{\boldsymbol{\psi}}_\mathrm{R}^* \right\}
     :label: tauM
 
@@ -58,28 +58,28 @@ The flux estimate in stator coordinates can be expressed using the polar form, :
 
 .. math::
     \frac{\mathrm{d}\hat{\vartheta}_\mathrm{s}}{\mathrm{d} t} = \omega_\mathrm{s}
-    = \frac{\mathrm{Im} \{ \boldsymbol{v}' + \boldsymbol{k}_1(\hat{\boldsymbol{v}} - \boldsymbol{v}') + \boldsymbol{k}_2(\hat{\boldsymbol{v}} - \boldsymbol{v}')^* \} }{\hat{\psi}_\mathrm{R} + L_\sigma \mathrm{Re}\{(1 - \boldsymbol{k}_1)\boldsymbol{i}_\mathrm{s} + \boldsymbol{k}_2 \boldsymbol{i}_\mathrm{s}^* \}}
+    = \frac{\mathrm{Im} \{ \hat{\boldsymbol{v}}_\mathrm{s}' + \boldsymbol{k}_1(\hat{\boldsymbol{v}} - \hat{\boldsymbol{v}}_\mathrm{s}') + \boldsymbol{k}_2(\hat{\boldsymbol{v}} - \hat{\boldsymbol{v}}_\mathrm{s}')^* \} }{\hat{\psi}_\mathrm{R} + \hat L_\sigma \mathrm{Re}\{(1 - \boldsymbol{k}_1)\boldsymbol{i}_\mathrm{s} + \boldsymbol{k}_2 \boldsymbol{i}_\mathrm{s}^*\}}
     :label: hatws
 
 where
 
 .. math::
-    \boldsymbol{v}' = \boldsymbol{u}_\mathrm{s} - R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - L_\sigma \frac{\mathrm{d} \boldsymbol{i}_\mathrm{s}}{\mathrm{d} t}
+    \hat{\boldsymbol{v}}_\mathrm{s}' = \boldsymbol{u}_\mathrm{s} - \hat R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - \hat L_\sigma \frac{\mathrm{d} \boldsymbol{i}_\mathrm{s}}{\mathrm{d} t}
     :label: vp
 
 The flux magnitude dynamics are
 
 .. math::
     \frac{\mathrm{d} \hat{\psi}_\mathrm{R}}{\mathrm{d} t}
-    = \mathrm{Re}\{ \boldsymbol{v} + \boldsymbol{k}_1(\hat{\boldsymbol{v}} - \boldsymbol{v}) + \boldsymbol{k}_2(\hat{\boldsymbol{v}} - \boldsymbol{v})^* \}
+    = \mathrm{Re}\{ \hat{\boldsymbol{v}}_\mathrm{s} + \boldsymbol{k}_1(\hat{\boldsymbol{v}} - \hat{\boldsymbol{v}}_\mathrm{s}) + \boldsymbol{k}_2(\hat{\boldsymbol{v}} - \hat{\boldsymbol{v}}_\mathrm{s})^* \}
     :label: dhatpsiR_abs
 
-Notice that the right-hand side of :eq:`hatws` is independent of :math:`\omega_\mathrm{s}`. Futhermore, in these coordinates, the condition :eq:`inherently` for an inherently sensorless observer reduces to :math:`\boldsymbol{k}_2 = \boldsymbol{k}_1`. This observer structure is implemented in the :class:`motulator.drive.control.im.Observer`, where simple forward-Euler discretization is used.
+Notice that the right-hand side of :eq:`hatws` is independent of :math:`\omega_\mathrm{s}`. Furthermore, in these coordinates, the condition :eq:`inherently` for an inherently sensorless observer reduces to :math:`\boldsymbol{k}_2 = \boldsymbol{k}_1`. This observer structure is implemented in the :class:`motulator.drive.control.im.Observer`, where simple forward-Euler discretization is used.
 
 Gain Selection
 ^^^^^^^^^^^^^^
 
-The estimation-error dynamics are obtained by subtracting :eq:`dhatpsiR` from :eq:`dpsiR`. The resulting system can be linearized for analysis and gain selection purposes. Using the rotor speed as an example, the small-signal deviation about the operating point is :math:`\Delta \omega_\mathrm{m} = \omega_\mathrm{m} - \omega_\mathrm{m0}`, where the subscript 0 refers to the operating point. Linearization of the estimation-error dynamics leads to [#Hin2010]_
+The estimation-error dynamics are obtained by subtracting :eq:`dhatpsiR` from :eq:`dpsiR`. The resulting system can be linearized for analysis and gain selection purposes. Accurate parameter estimates are assumed in the analysis. Using the rotor speed as an example, the small-signal deviation about the operating point is :math:`\Delta \omega_\mathrm{m} = \omega_\mathrm{m} - \omega_\mathrm{m0}`, where the subscript 0 refers to the operating point. Linearization of the estimation-error dynamics leads to [#Hin2010]_
 
 .. math::
 	\frac{\mathrm{d} \Delta\tilde{\boldsymbol{\psi}}_\mathrm{R}}{\mathrm{d} t} =  \boldsymbol{k}_1\Delta \tilde{\boldsymbol{v}} + \boldsymbol{k}_2\Delta \tilde{\boldsymbol{v}}^* - \mathrm{j}\omega_\mathrm{s0}\Delta\tilde{\boldsymbol{\psi}}_\mathrm{R}
@@ -103,7 +103,7 @@ Here, :math:`\boldsymbol{k}_2 = 0` and :math:`\Delta\tilde{\omega}_\mathrm{m} =
 It can be noticed that the closed-loop pole could be arbitrarily placed via the gain :math:`\boldsymbol{k}_1`. Well-damped estimation-error dynamics can be obtained, e.g., by choosing
 
 .. math::
-    \boldsymbol{k}_1 = 1 + \frac{g |\omega_\mathrm{m}|}{\alpha - \mathrm{j}\omega_\mathrm{m}}
+    \boldsymbol{k}_1 = 1 + \frac{g |\omega_\mathrm{m}|}{\hat \alpha - \mathrm{j}\omega_\mathrm{m}}
     :label: k1_sensored
 
 where :math:`g` is a unitless positive design parameter. The corresponding pole is located at :math:`s = -\alpha - g |\omega_\mathrm{m0}| - \mathrm{j}\omega_\mathrm{r0}`, where :math:`\omega_\mathrm{r0} = \omega_\mathrm{s0} - \omega_\mathrm{m0}` is the operating-point slip angular frequency.
@@ -131,10 +131,10 @@ yields an inherently sensorless observer, i.e., the rotor speed estimate :math:`
 where the gain components correspond to :math:`\boldsymbol{k}_1 = k_\mathrm{d} + \mathrm{j}k_\mathrm{q}`. It can be seen that the dynamics of the rotor speed are decoupled from the flux-estimation error dynamics. The decay rate :math:`\sigma` be assigned by choosing
 
 .. math::
-    \boldsymbol{k}_1 = \frac{\sigma}{\alpha - \mathrm{j}\hat\omega_\mathrm{m}}
+    \boldsymbol{k}_1 = \frac{\sigma}{\hat \alpha - \mathrm{j}\hat\omega_\mathrm{m}}
     :label: k1_sensorless
 
-which results in the characteristic polynomial :math:`D(s)=s^2 + 2\sigma s + \omega_\mathrm{s0}^2`. The decay rate can be selected as :math:`\sigma = \alpha/2 + \zeta_\infty|\hat{\omega}_\mathrm{m}|`, where :math:`\zeta_\infty` is the desired damping ratio at high speed. At zero stator frequency :math:`\omega_\mathrm{s0} = 0`, the poles are located at :math:`s = 0` and :math:`s = -\alpha`, which allows stable magnetizing and starting the machine.
+which results in the characteristic polynomial :math:`D(s)=s^2 + 2\sigma s + \omega_\mathrm{s0}^2`. The decay rate can be selected as :math:`\sigma = \hat \alpha/2 + \zeta_\infty|\hat{\omega}_\mathrm{m}|`, where :math:`\zeta_\infty` is the desired damping ratio at high speeds. At zero stator frequency :math:`\omega_\mathrm{s0} = 0`, the poles are located at :math:`s = 0` and :math:`s = -\alpha`, which allows stable magnetizing and starting the machine.
 
 .. rubric:: References
 
@@ -179,13 +179,13 @@ The observer is assumed to operate in estimated rotor coordinates, whose d-axis
 Since the stator current is measured, the observer is fundamentally corrected by means of the current estimation error. However, due to the saliency, it is more convenient to scale the current estimation error by the stator inductance, resulting in the flux linkage error
 
 .. math::
-	\boldsymbol{e} = \psi_\mathrm{f} + L_\mathrm{d} \mathrm{Re}\{ \boldsymbol{i}_\mathrm{s}\} + \mathrm{j} L_\mathrm{q} \mathrm{Im}\{\boldsymbol{i}_\mathrm{s} \} - \hat{\boldsymbol{\psi}}_\mathrm{s}
+	\boldsymbol{e} = \hat \psi_\mathrm{f} + \hat L_\mathrm{d} \mathrm{Re}\{ \boldsymbol{i}_\mathrm{s}\} + \mathrm{j} \hat L_\mathrm{q} \mathrm{Im}\{\boldsymbol{i}_\mathrm{s} \} - \hat{\boldsymbol{\psi}}_\mathrm{s}
     :label: e
 
 where :math:`\hat{\boldsymbol{\psi}}_\mathrm{s}` is the stator flux estimate. The flux linkage is estimated by
 
 .. math::
-    \frac{\mathrm{d} \hat{\boldsymbol{\psi}}_\mathrm{s}}{\mathrm{d} t} = \boldsymbol{u}_\mathrm{s} - R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - \mathrm{j}\omega_\mathrm{s}\hat{\boldsymbol{\psi}}_\mathrm{s} + \boldsymbol{k}_1 \boldsymbol{e} + \boldsymbol{k}_2 \boldsymbol{e}^*
+    \frac{\mathrm{d} \hat{\boldsymbol{\psi}}_\mathrm{s}}{\mathrm{d} t} = \boldsymbol{u}_\mathrm{s} - \hat R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - \mathrm{j}\omega_\mathrm{s}\hat{\boldsymbol{\psi}}_\mathrm{s} + \boldsymbol{k}_1 \boldsymbol{e} + \boldsymbol{k}_2 \boldsymbol{e}^*
     :label: sm_flux_observer
 
 where :math:`\boldsymbol{k}_1` and :math:`\boldsymbol{k}_2` are gains (complex in a general case), the estimates are marked with the hat, and :math:`^*` marks the complex conjugate.
@@ -240,7 +240,7 @@ where :math:`\sigma` is the desired decay rate of the flux estimation error and
 allows to decouple the flux-estimation error dynamics from the rotor-position dynamics. By default, the decay rate is scheduled as
 
 .. math::
-    \sigma = \frac{R_\mathrm{s}}{4}\left(\frac{1}{L_\mathrm{d}} + \frac{1}{L_\mathrm{q}}\right) + \zeta_\infty |\hat{\omega}_\mathrm{m} |
+    \sigma = \frac{\hat R_\mathrm{s}}{4}\left(\frac{1}{\hat L_\mathrm{d}} + \frac{1}{\hat L_\mathrm{q}}\right) + \zeta_\infty |\hat{\omega}_\mathrm{m} |
     :label: sigma_sensorless
 
 where :math:`\zeta_\infty` is the desired damping ratio at high speed. At zero speed, :eq:`sigma_sensorless` places one pole at :math:`s = 0` and another at :math:`s = -(R_\mathrm{s}/2)(1/L_\mathrm{d} + 1/L_\mathrm{q})`.
diff --git a/docs/_sources/control/grid/current_ctrl.rst b/docs/_sources/control/grid/current_ctrl.rst
index 8d279e8e..9306a3b7 100644
--- a/docs/_sources/control/grid/current_ctrl.rst
+++ b/docs/_sources/control/grid/current_ctrl.rst
@@ -1,7 +1,7 @@
 Current Control
 ===============
 
-Synchronous-frame two-degrees-of-freedom (2DOF) proportional-integral (PI) current control can be used in grid converters [#Har2015]_. This control structure allows to compensate for the cross-coupling originating from rotating coordinates as well as to improve disturbance rejection. A 2DOF PI current controller is available in the :class:`motulator.grid.control.CurrentController` class, whose base class is :class:`motulator.common.control.ComplexPIController`. 
+Synchronous-frame two-degrees-of-freedom (2DOF) proportional-integral (PI) current control can be used in grid converters [#Har2015]_. This control structure allows to compensate for the cross-coupling originating from rotating coordinates as well as to improve disturbance rejection. A 2DOF PI current controller is available in the :class:`motulator.grid.control.CurrentController` class, whose base class is :class:`motulator.common.control.ComplexPIController`.
 
 .. note:: This controller design assumes an L filter, but it can also be applied with LCL filters (see the :doc:`/grid_examples/grid_following/plot_gfl_lcl_10kva` example). If LCL-resonance damping and very low sampling frequencies are needed, the controller could be designed directly in the discrete-time domain taking the LCL filter dynamics into account [#Rah2021]_.
 
@@ -12,13 +12,13 @@ The design of synchronous-frame 2DOF PI current control is considered in the con
 
 .. math::
 	\frac{\mathrm{d} \boldsymbol{u}_\mathrm{i}}{\mathrm{d} t} &= (\boldsymbol{k}_\mathrm{i} + \mathrm{j}\omega_\mathrm{c}\boldsymbol{k}_\mathrm{t} )\left(\boldsymbol{i}_\mathrm{c,ref} - \boldsymbol{i}_\mathrm{c}\right) \\
-    \boldsymbol{u}_\mathrm{c,ref} &= \boldsymbol{k}_\mathrm{t}\boldsymbol{i}_\mathrm{c,ref} - \boldsymbol{k}_\mathrm{p}\boldsymbol{i}_\mathrm{c} + \boldsymbol{u}_\mathrm{i} 
+    \boldsymbol{u}_\mathrm{c,ref} &= \boldsymbol{k}_\mathrm{t}\boldsymbol{i}_\mathrm{c,ref} - \boldsymbol{k}_\mathrm{p}\boldsymbol{i}_\mathrm{c} + \boldsymbol{u}_\mathrm{i}
     :label: grid_cc
 
-where :math:`\boldsymbol{u}_\mathrm{c,ref}` is the output of the controller, i.e., the converter voltage reference, :math:`\boldsymbol{i}_\mathrm{c}` is the measured converter current, :math:`\boldsymbol{i}_\mathrm{c,ref}` is the converter current reference, :math:`\boldsymbol{u}_\mathrm{i}` is the the integral state, and :math:`\omega_\mathrm{c}` is the angular speed of the coordinate system. Furthermore, :math:`\boldsymbol{k}_\mathrm{t}` is the reference-feedforward gain, :math:`\boldsymbol{k}_\mathrm{p}` is the proportional gain, and :math:`\boldsymbol{k}_\mathrm{i}` is the integral gain. 
+where :math:`\boldsymbol{u}_\mathrm{c,ref}` is the output of the controller, i.e., the converter voltage reference, :math:`\boldsymbol{i}_\mathrm{c}` is the measured converter current, :math:`\boldsymbol{i}_\mathrm{c,ref}` is the converter current reference, :math:`\boldsymbol{u}_\mathrm{i}` is the the integral state, and :math:`\omega_\mathrm{c}` is the angular speed of the coordinate system. Furthermore, :math:`\boldsymbol{k}_\mathrm{t}` is the reference-feedforward gain, :math:`\boldsymbol{k}_\mathrm{p}` is the proportional gain, and :math:`\boldsymbol{k}_\mathrm{i}` is the integral gain.
 
-Closed-Loop System 
-^^^^^^^^^^^^^^^^^^
+Closed-Loop System
+------------------
 
 Consider the grid model in synchronous coordinates
 
@@ -34,23 +34,23 @@ where :math:`\boldsymbol{u}_\mathrm{c}` is the converter output voltage, :math:`
 The disturbance rejection depends on the closed-loop admittance
 
 .. math::
-    \boldsymbol{Y}_\mathrm{c}(s) = \frac{s}{L s^2 + (\boldsymbol{k}_\mathrm{p} + \mathrm{j}\omega_\mathrm{c} L) s + \boldsymbol{k}_\mathrm{i} + \mathrm{j}\omega_\mathrm{c} \boldsymbol{k}_\mathrm{t}} 
+    \boldsymbol{Y}_\mathrm{c}(s) = \frac{s}{L s^2 + (\boldsymbol{k}_\mathrm{p} + \mathrm{j}\omega_\mathrm{c} L) s + \boldsymbol{k}_\mathrm{i} + \mathrm{j}\omega_\mathrm{c} \boldsymbol{k}_\mathrm{t}}
     :label: Yc_grid
 
 The closed-loop poles can be arbitrarily placed by means of the gains. The reference-tracking transfer function is
 
 .. math::
-	\boldsymbol{G}_\mathrm{c}(s) = \frac{(s + \mathrm{j}\omega_\mathrm{c}) \boldsymbol{k}_\mathrm{t} + \boldsymbol{k}_\mathrm{i} }{L s^2 + (\boldsymbol{k}_\mathrm{p} + \mathrm{j}\omega_\mathrm{c} L) s + \boldsymbol{k}_\mathrm{i} + \mathrm{j}\omega_\mathrm{c} \boldsymbol{k}_\mathrm{t}}     
+	\boldsymbol{G}_\mathrm{c}(s) = \frac{(s + \mathrm{j}\omega_\mathrm{c}) \boldsymbol{k}_\mathrm{t} + \boldsymbol{k}_\mathrm{i} }{L s^2 + (\boldsymbol{k}_\mathrm{p} + \mathrm{j}\omega_\mathrm{c} L) s + \boldsymbol{k}_\mathrm{i} + \mathrm{j}\omega_\mathrm{c} \boldsymbol{k}_\mathrm{t}}
     :label: Gc_grid
 
 whose zero can be placed by means of the reference-feedforward gain :math:`\boldsymbol{k}_\mathrm{t}`.
 
 Gain Selection
-^^^^^^^^^^^^^^
+--------------
 
 Consider the gains
 
-.. math::                
+.. math::
     \boldsymbol{k}_\mathrm{p} = 2\alpha_\mathrm{c} \hat L \qquad\qquad
     \boldsymbol{k}_\mathrm{i} = \alpha_\mathrm{c}^2\hat L  \qquad \qquad
     \boldsymbol{k}_\mathrm{t} = \alpha_\mathrm{c} \hat L
@@ -60,9 +60,9 @@ where :math:`\hat L` is the inductance estimate. Assuming accurate parameter est
 .. math::
     \boldsymbol{Y}_\mathrm{c}(s) = \frac{s}{L (s + \alpha_\mathrm{c})(s + \alpha_\mathrm{c} + \mathrm{j}\omega_\mathrm{c})}
     \qquad\qquad
-    \boldsymbol{G}_\mathrm{c}(s) = \frac{\alpha_\mathrm{c}}{s + \alpha_\mathrm{c}} 
+    \boldsymbol{G}_\mathrm{c}(s) = \frac{\alpha_\mathrm{c}}{s + \alpha_\mathrm{c}}
 
-It can be seen that this design results in the first-order reference-tracking dynamics. Furthermore, one pole is placed at the real axis at :math:`s=-\alpha_\mathrm{c}` and another pole at :math:`s= -\alpha_\mathrm{c} - \mathrm{j}\omega_\mathrm{c}`. This gain selection is used in the :class:`motulator.grid.control.CurrentController` class. 
+It can be seen that this design results in the first-order reference-tracking dynamics. Furthermore, one pole is placed at the real axis at :math:`s=-\alpha_\mathrm{c}` and another pole at :math:`s= -\alpha_\mathrm{c} - \mathrm{j}\omega_\mathrm{c}`. This gain selection is used in the :class:`motulator.grid.control.CurrentController` class.
 
 The converter output voltage is limited in practice due to the limited DC-bus voltage of the converter. Consequently, the realized (limited) voltage reference is
 
@@ -80,6 +80,3 @@ where :math:`\mathrm{sat}(\cdot)` is the saturation function. The limited voltag
 .. [#Har1998] Harnefors, Nee, "Model-based current control of AC machines using the internal model control method," IEEE Trans. Ind. Appl., 1998, https://doi.org/10.1109/28.658735
 
 .. [#Bri1999] Briz del Blanco, Degner, Lorenz, “Dynamic analysis of current regulators for AC motors using complex vectors,” IEEE Trans. Ind. Appl., 1999, https://doi.org/10.1109/28.806058
-
-
-
diff --git a/docs/_sources/control/grid/dc_voltage_ctrl.rst b/docs/_sources/control/grid/dc_voltage_ctrl.rst
index 23bc2ed6..d9cbd2cf 100644
--- a/docs/_sources/control/grid/dc_voltage_ctrl.rst
+++ b/docs/_sources/control/grid/dc_voltage_ctrl.rst
@@ -1,7 +1,7 @@
 DC-Bus Voltage Control
 ======================
 
-A PI controller for the DC-bus voltage is implemented in the :class:`motulator.grid.control.DCBusVoltageController` class, whose base class is :class:`motulator.common.control.PIController`. In the following, the tuning of the DC-bus voltage controller is considered in the continuous-time domain. 
+A PI controller for the DC-bus voltage is implemented in the :class:`motulator.grid.control.DCBusVoltageController` class, whose base class is :class:`motulator.common.control.PIController`. In the following, the tuning of the DC-bus voltage controller is considered in the continuous-time domain.
 
 Controller Structure
 --------------------
@@ -10,16 +10,16 @@ In order to have a linear closed-loop system, it is convenient to use the DC-bus
 
 .. math::
 	\hat W_\mathrm{dc} = \frac{1}{2}\hat{C}_\mathrm{dc} u_\mathrm{dc}^2 \qquad
-    W_\mathrm{dc,ref} = \frac{1}{2}\hat{C}_\mathrm{dc} u_\mathrm{dc,ref}^2 
-    :label: dc_cap_energy 
+    W_\mathrm{dc,ref} = \frac{1}{2}\hat{C}_\mathrm{dc} u_\mathrm{dc,ref}^2
+    :label: dc_cap_energy
 
-where :math:`\hat{C}_\mathrm{dc}` is the DC-bus capacitance estimate, :math:`u_\mathrm{dc}` is the measured DC-bus voltage, and :math:`u_\mathrm{dc,ref}` is the DC-bus voltage reference. Using these variables, the output power reference :math:`p_\mathrm{c,ref}` for the converter is obtained from a PI controller, 
+where :math:`\hat{C}_\mathrm{dc}` is the DC-bus capacitance estimate, :math:`u_\mathrm{dc}` is the measured DC-bus voltage, and :math:`u_\mathrm{dc,ref}` is the DC-bus voltage reference. Using these variables, the output power reference :math:`p_\mathrm{c,ref}` for the converter is obtained from a PI controller,
 
 .. math::
     p_\mathrm{c,ref} = -k_\mathrm{p} (W_\mathrm{dc,ref} - \hat{W}_\mathrm{dc}) - \int k_\mathrm{i} (W_\mathrm{dc,ref} - \hat{W}_\mathrm{dc}) \mathrm{d} t
     :label: dc_voltage_ctrl
 
-where :math:`k_\mathrm{p}` is the proportional gain and :math:`k_\mathrm{i}` is the integral gain. The negative signs appear since the positive converter output power decreases the DC-bus capacitor energy, see :eq:`capacitor`. 
+where :math:`k_\mathrm{p}` is the proportional gain and :math:`k_\mathrm{i}` is the integral gain. The negative signs appear since the positive converter output power decreases the DC-bus capacitor energy, see :eq:`capacitor`.
 
 Closed-Loop System
 ------------------
@@ -38,7 +38,7 @@ In the Laplace domain, the closed-loop system resulting from :eq:`dc_voltage_ctr
     W_\mathrm{dc}(s) = \frac{k_\mathrm{p} s + k_\mathrm{i}}{s^2 + k_\mathrm{p} s + k_\mathrm{i}} W_\mathrm{dc,ref}(s) + \frac{s}{s^2 + k_\mathrm{p} s + k_\mathrm{i}} p_\mathrm{dc}(s)
     :label: dc_voltage_ctrl_closed_loop
 
-where it can be seen that :math:`W_\mathrm{dc} = W_\mathrm{dc,ref}` holds in the steady state. Furthermore, it can be shown that also :math:`u_\mathrm{dc} = u_\mathrm{dc,ref}` holds in the steady state (independently of the errors in the capacitance estimate :math:`\hat{C}`, since both reference and measured values in :eq:`dc_cap_energy` are scaled by the same estimate). 
+where it can be seen that :math:`W_\mathrm{dc} = W_\mathrm{dc,ref}` holds in the steady state. Furthermore, it can be shown that also :math:`u_\mathrm{dc} = u_\mathrm{dc,ref}` holds in the steady state (independently of the errors in the capacitance estimate :math:`\hat{C}`, since both reference and measured values in :eq:`dc_cap_energy` are scaled by the same estimate).
 
 Gain Selection
 --------------
@@ -47,9 +47,9 @@ Based on :eq:`dc_voltage_ctrl_closed_loop`, the gain selection
 
 .. math::
     k_\mathrm{p} = 2\alpha_\mathrm{dc} \qquad
-    k_\mathrm{i} = \alpha_\mathrm{dc}^2  
+    k_\mathrm{i} = \alpha_\mathrm{dc}^2
 
-results in the double real pole at :math:`s = -\alpha_\mathrm{dc}`. The closed-loop bandwidth is approximately :math:`\alpha_\mathrm{dc}`. 
+results in the double real pole at :math:`s = -\alpha_\mathrm{dc}`. The closed-loop bandwidth is approximately :math:`\alpha_\mathrm{dc}`.
 
 .. rubric:: References
 
diff --git a/docs/_sources/control/grid/grid-forming.rst b/docs/_sources/control/grid/grid-forming.rst
new file mode 100644
index 00000000..939e90c6
--- /dev/null
+++ b/docs/_sources/control/grid/grid-forming.rst
@@ -0,0 +1,92 @@
+Disturbance-Observer-Based Grid-Forming Control
+===============================================
+
+In these notes, disturbance-observer-based grid-forming control is discussed [#Nur2024]_. This control method is available in the :class:`motulator.grid.control.ObserverBasedGridFormingControl` class. As can be realized by comparing the examples :doc:`/grid_examples/grid_forming/plot_gfm_rfpsc_13kva` and :doc:`/grid_examples/grid_forming/plot_gfm_obs_13kva`, the observer-based grid-forming control can be configured to behave practically identically with reference-feedforward power-synchronization control (PSC) [#Har2020]_. Compared to reference-feedforward PSC, disturbance-observer-based grid-forming control is easier to analyze and extend with different operating modes (including grid-following and transparent current-control modes). The order of these two control methods is the same and their computational complexity is comparable.
+
+System Model
+------------
+
+First, the system is modeled in general coordinates, whose angle with respect to the stationary coordinates is :math:`\vartheta_\mathrm{c}` and the angular speed is :math:`\omega_\mathrm{c} = \mathrm{d}\vartheta_\mathrm{c}/\mathrm{d} t`. The dynamics of the inductor current :math:`\boldsymbol{i}_\mathrm{c}` and the grid voltage :math:`\boldsymbol{u}_\mathrm{g}` are modeled as
+
+.. math::
+    L\frac{\mathrm{d} \boldsymbol{i}_\mathrm{c}}{\mathrm{d} t} &= \boldsymbol{u}_\mathrm{c} - \boldsymbol{u}_\mathrm{g} - \mathrm{j}\omega_\mathrm{c} L \boldsymbol{i}_\mathrm{c} \\
+    \frac{\mathrm{d}\boldsymbol{u}_\mathrm{g}}{\mathrm{d} t} &= \mathrm{j}(\omega_\mathrm{g} - \omega_\mathrm{c})\boldsymbol{u}_\mathrm{g}
+    :label: system_model1
+
+where :math:`L` is the inductance and :math:`\omega_\mathrm{g}` is the grid angular frequency. The grid voltage :math:`\boldsymbol{u}_\mathrm{g}` is modeled as a disturbance; this same disturbance model for the grid voltage is also used in the development of :doc:`/control/grid/pll`. The active power fed to the grid is nonlinear in the state variables
+
+.. math::
+    p_\mathrm{g} = \frac{3}{2}\mathrm{Re}\{\boldsymbol{u}_\mathrm{g}\boldsymbol{i}_\mathrm{c}^*\}
+    :label: power1
+
+For the purpose of grid-forming control, we may define the quasi-static converter voltage as an output variable as a function of the state variables,
+
+.. math::
+    \boldsymbol{v}_\mathrm{c} = \boldsymbol{u}_\mathrm{g} + \mathrm{j}\omega_\mathrm{g} L \boldsymbol{i}_\mathrm{c}
+    :label: quasi_static_converter_voltage
+
+This voltage is controllable (unlike the grid voltage which is the disturbance) and it equals the converter output voltage :math:`\boldsymbol{u}_\mathrm{c}` in the steady state. The disturbance-observer-based PSC regulates the voltage magnitude :math:`|\boldsymbol{v}_\mathrm{c}|` and the power :math:`p_\mathrm{g}` fed to grid.
+
+Control System
+--------------
+
+The control system consists of a disturbance observer and a control law. The disturbance observer estimates the quasi-static converter voltage and the power fed to the grid. The control law generates the converter voltage reference based on the estimated grid voltage and power. The disturbance observer also provides the integral action for the control law. This control structure allows seamless switching between grid-forming and grid-following modes [#Nur2024]_. For simplicity, only the grid-forming mode is considered in the following.
+
+Disturbance Observer
+^^^^^^^^^^^^^^^^^^^^
+
+Based on :eq:`system_model1`--:eq:`quasi_static_converter_voltage`, a disturbance observer for the grid voltage can be formed [#Nur2024]_, [#Kuk2021]_, [#Fra1997]_
+
+.. math::
+    \frac{\mathrm{d} \hat{\boldsymbol{u}}_\mathrm{g}}{\mathrm{d} t} &= \mathrm{j} (\hat{\omega}_\mathrm{g} - \omega_\mathrm{c})\hat{\boldsymbol{u}}_\mathrm{g} + \alpha_\mathrm{o}\left(\boldsymbol{u}_\mathrm{c} - \hat L \frac{\mathrm{d} \boldsymbol{i}_\mathrm{c}}{\mathrm{d} t} - \mathrm{j} \omega_\mathrm{c} \hat L \boldsymbol{i}_\mathrm{c} - \hat{\boldsymbol{u}}_\mathrm{g} \right) \\
+    \hat{\boldsymbol{v}}_\mathrm{c} &= \hat{\boldsymbol{u}}_\mathrm{g} + \mathrm{j}\hat{\omega}_\mathrm{g} \hat L \boldsymbol{i}_\mathrm{c} \\
+    \hat p_\mathrm{g} &= \frac{3}{2}\mathrm{Re}\{\hat{\boldsymbol{v}}_\mathrm{c}\boldsymbol{i}_\mathrm{c}^*\}
+    :label: disturbance_observer_gfm
+
+where :math:`\hat \omega_\mathrm{g}` is the nominal grid angular frequency and estimates are marked with hat.
+
+.. note:: A conventional phase-locked loop (PLL) can be expressed in the same disturbance observer framework, see the :doc:`/control/grid/pll` notes. It can be realized that the measured grid voltage :math:`\boldsymbol{u}_\mathrm{g}` used in the conventional PLL is replaced by its converter-voltage-based estimate :math:`\boldsymbol{u}_\mathrm{c} - \hat L (\mathrm{d} \boldsymbol{i}_\mathrm{c}/\mathrm{d} t) - \mathrm{j} \omega_\mathrm{c} \hat L \boldsymbol{i}_\mathrm{c}` in the disturbance observer :eq:`disturbance_observer_gfm`.
+
+Control Law
+^^^^^^^^^^^
+
+A nonlinear state feedback law is used
+
+.. math::
+    \boldsymbol{u}_\mathrm{c,ref} = \hat{\boldsymbol{v}}_\mathrm{c} + \boldsymbol{k}_\mathrm{p} (p_\mathrm{g,ref} - \hat p_\mathrm{g}) + \boldsymbol{k}_\mathrm{v} (v_\mathrm{c,ref} - |\hat{\boldsymbol{v}}_\mathrm{c}|)
+    :label: control_law_gfm
+
+where :math:`p_\mathrm{g,ref}` is the active power reference, :math:`v_\mathrm{c,ref}` is the converter voltage magnitude reference, and :math:`\boldsymbol{k}_\mathrm{p}` and :math:`\boldsymbol{k}_\mathrm{v}` are the complex gains for the active-power and converter-voltage-magnitude channels, respectively. To ensure robust operation, similar to the reference-feedforward PSC, the complex gains can be selected as
+
+.. math::
+    \boldsymbol{k}_\mathrm{p} = \frac{R_\mathrm{a}}{v_\mathrm{c,ref}} \frac{\hat{\boldsymbol{v}}_\mathrm{c}}{|\hat{\boldsymbol{v}}_\mathrm{c}|} \qquad
+    \boldsymbol{k}_\mathrm{v} = (1 - \mathrm{j} k_\mathrm{v}) \frac{\hat{\boldsymbol{v}}_\mathrm{c}}{|\hat{\boldsymbol{v}}_\mathrm{c}|}
+    :label: gain_selection_gfm
+
+where the gains :math:`R_\mathrm{a} = 0.2` p.u. and :math:`k_\mathrm{v} = \alpha_\mathrm{o}/\omega_\mathrm{g}` can be used.
+
+Implementation Aspects
+^^^^^^^^^^^^^^^^^^^^^^
+
+To avoid the derivate on the right-hand side of :eq:`disturbance_observer_gfm`, a new state variable :math:`\hat{\boldsymbol{u}}_\mathrm{g}' = \hat{\boldsymbol{u}}_\mathrm{g} + \alpha_\mathrm{o} \hat L \boldsymbol{i}_\mathrm{c}` can be introduced [#Fra1997]_. Furthermore, the coordinate system for the implementation can be chosen freely. The simplest choice is to use the nominal grid frequency as the coordinate system frequency, :math:`\omega_\mathrm{c} = \hat \omega_\mathrm{g}`. Using these design choices, the whole control system consisting of the disturbance observer :eq:`disturbance_observer_gfm` and the control law :eq:`control_law_gfm` in the state-space form reduces to [#Nur2024]_
+
+.. math::
+    \frac{\mathrm{d} \hat{\boldsymbol{u}}'_\mathrm{g}}{\mathrm{d} t} &= \alpha_\mathrm{o} (\boldsymbol{u}_\mathrm{c,ref} - \hat{\boldsymbol{v}}_\mathrm{c} ) \\
+    \hat{\boldsymbol{v}}_\mathrm{c} &= \hat{\boldsymbol{u}}_\mathrm{g}' - (\alpha_\mathrm{o} - \mathrm{j}\hat{\omega}_\mathrm{g}) \hat L \boldsymbol{i}_\mathrm{c} \\
+    \hat p_\mathrm{g} &= \frac{3}{2}\mathrm{Re}\{\hat{\boldsymbol{v}}_\mathrm{c}\boldsymbol{i}_\mathrm{c}^*\} \\
+    \boldsymbol{u}_\mathrm{c,ref} &= \hat{\boldsymbol{v}}_\mathrm{c} + \boldsymbol{k}_\mathrm{p} (p_\mathrm{g,ref} - \hat p_\mathrm{g}) + \boldsymbol{k}_\mathrm{v} (v_\mathrm{c,ref} - |\hat{\boldsymbol{v}}_\mathrm{c}|)
+    :label: control_system_gfm
+
+where the gains can be selected according to :eq:`gain_selection_gfm` and the converter voltage appearing in the observer has been replaced with its reference. Various control modes could be easily incorporated into the control system :eq:`control_system_gfm`, simply by changing the feedback correction terms of the control law [#Nur2024]_. The switching between the modes is seamless since the control law does not have memory, but the integral action is provided by the disturbance observer (in addition to synchronization).
+
+The control system implemented in the :class:`motulator.grid.control.ObserverBasedGridFormingControl` class corresponds to :eq:`control_system_gfm`. In the example implementation, a transparent current-control mode is implemented. In the grid-forming mode, the observer bandwidth :math:`\alpha_\mathrm{o} = 1` p.u. can be used. Furthermore, the inductance estimate can be set close to the lowest expected inductance value, e.g., :math:`\hat L = 0.15` p.u. Using this configuration, the robust performance from strong grids to very weak grids can be achieved. This grid-forming control method can also be used with LCL filters, similarly to reference-feedforward PSC.
+
+.. rubric:: References
+
+.. [#Nur2024] Nurminen, Mourouvin, Hinkkanen, Kukkola, "Multifunctional grid-forming converter control based on a disturbance observer, "IEEE Trans. Power Electron., 2024, https://doi.org/10.1109/TPEL.2024.3433503
+
+.. [#Har2020] Harnefors, Rahman, Hinkkanen, Routimo, "Reference-feedforward power-synchronization control," IEEE Trans. Power Electron., 2020, https://doi.org/10.1109/TPEL.2020.2970991
+
+.. [#Kuk2021] Kukkola, Routimo, Hinkkanen, Harnefors, "A voltage-sensorless controller for grid converters," IEEE PES ISGT Europe, 2021, https://doi.org/10.1109/ISGTEurope52324.2021.9640206
+
+.. [#Fra1997] Franklin, Powell, Workman, "Digital Control of Dynamic Systems," 3rd ed., Menlo Park, CA: Addison-Wesley, 1997
diff --git a/docs/_sources/control/grid/index.rst b/docs/_sources/control/grid/index.rst
index c91e97d3..4e4a3f0e 100644
--- a/docs/_sources/control/grid/index.rst
+++ b/docs/_sources/control/grid/index.rst
@@ -2,7 +2,7 @@
 Grid Converters
 ***************
 
-Design notes for selected control methods for grid-connected converters are provided in this section. The aim of these notes is to link the implemented methods to the theory and to provide a reference for the implementation. Further details are available in the references provided. 
+Design notes for selected control methods for grid-connected converters are provided in this section. The aim of these notes is to link the implemented methods to the theory and to provide a reference for the implementation. Further details are available in the references provided.
 
 .. toctree::
    :titlesonly:
@@ -10,4 +10,5 @@ Design notes for selected control methods for grid-connected converters are prov
 
    dc_voltage_ctrl
    current_ctrl
-   synchronization
+   Phase-Locked Loop 
+   Grid-Forming Control 
diff --git a/docs/_sources/control/grid/pll.rst b/docs/_sources/control/grid/pll.rst
new file mode 100644
index 00000000..31986724
--- /dev/null
+++ b/docs/_sources/control/grid/pll.rst
@@ -0,0 +1,100 @@
+Phase-Locked Loop
+=================
+
+A phase-locked loop (PLL) is commonly used in grid-following converters to synchronize the converter output with the grid [#Kau1997]_. Here, we represent the PLL using the disturbance observer structure [#Fra1997]_, which may be simpler to extend than the classical PLL. Furthermore, this structure allows to see links to synchronization methods used in grid-forming converters, see [#Nur2024]_.
+
+Disturbance Model
+-----------------
+
+Consider the positive-sequence voltage at the point of common coupling (PCC) in general coordinates, rotating at the angular speed :math:`\omega_\mathrm{c}`. The dynamics of the PCC voltage :math:`\boldsymbol{u}_\mathrm{g}` can be modeled using a disturbance model as
+
+.. math::
+    \frac{\mathrm{d}\boldsymbol{u}_\mathrm{g}}{\mathrm{d}t} = \mathrm{j}(\omega_\mathrm{g} - \omega_\mathrm{c})\boldsymbol{u}_\mathrm{g}
+    :label: disturbance
+
+where :math:`\omega_\mathrm{g}` is the grid angular frequency. If needed, this disturbance model could be extended, e.g., with a negative-sequence component, allowing to design the PLL for unbalanced grids.
+
+.. note:: The disturbance model in :eq:`disturbance` can be equivalently expressed in polar form as
+
+    .. math::
+        \frac{\mathrm{d} u_\mathrm{g}}{\mathrm{d} t} &= 0 \\
+        \frac{\mathrm{d} \delta_\mathrm{g}}{\mathrm{d} t} &= \omega_\mathrm{g} - \omega_\mathrm{c} \\
+        \boldsymbol{u}_\mathrm{g} &= u_\mathrm{g} \mathrm{e}^{\mathrm{j}\delta_\mathrm{g}}
+
+    where :math:`\delta_\mathrm{g} = \vartheta_\mathrm{g} - \vartheta_\mathrm{c}` is the angle of the grid voltage in general coordinates and the last equation is the output equation.
+
+PLL in General Coordinates
+--------------------------
+
+Based on :eq:`disturbance`, the disturbance observer containing the regular PLL functionality can be formulated as
+
+.. math::
+    \frac{\mathrm{d}\hat{\boldsymbol{u}}_\mathrm{g}}{\mathrm{d}t} = \mathrm{j}(\hat{\omega}_\mathrm{g} - \omega_\mathrm{c})\hat{\boldsymbol{u}}_\mathrm{g} + \alpha_\mathrm{o} (\boldsymbol{u}_\mathrm{g} - \hat{\boldsymbol{u}}_\mathrm{g} )
+    :label: pll
+
+where :math:`\hat{\boldsymbol{u}}_\mathrm{g}` is the estimated PCC voltage, :math:`\hat{\omega}_\mathrm{g}` is the grid angular frequency estimate (either constant corresponding to the nominal value or dynamic from grid-frequency tracking), and :math:`\alpha_\mathrm{o}` is the bandwidth. If needed, the disturbance observer can be extended with the frequency tracking as
+
+.. math::
+    \frac{\mathrm{d}\hat{\omega}_\mathrm{g}}{\mathrm{d}t} = k_\omega\mathrm{Im}\left\{ \frac{\boldsymbol{u}_\mathrm{g} - \hat{\boldsymbol{u}}_\mathrm{g}}{\hat{\boldsymbol{u}}_\mathrm{g}} \right\}
+    :label: frequency_tracking
+
+where :math:`k_\omega` is the frequency-tracking gain. Notice that :eq:`pll` and :eq:`frequency_tracking` are both driven by the estimation error :math:`\boldsymbol{u}_\mathrm{g} - \hat{\boldsymbol{u}}_\mathrm{g}` of the PCC voltage.
+
+PLL in Estimated PCC Voltage Coordinates
+----------------------------------------
+
+The disturbance observer :eq:`pll` can be equivalently expressed in estimated PCC voltage coordinates, where :math:`\hat{\boldsymbol{u}}_\mathrm{g} = \hat{u}_\mathrm{g}` and :math:`\omega_\mathrm{c} = \mathrm{d} \hat{\vartheta}_\mathrm{g}/ \mathrm{d} t`, resulting in
+
+.. math::
+    \frac{\mathrm{d}\hat{u}_\mathrm{g}}{\mathrm{d}t} &= \alpha_\mathrm{o} \left(\mathrm{Re}\{\boldsymbol{u}_\mathrm{g}\} - \hat{u}_\mathrm{g} \right) \\
+    \frac{\mathrm{d} \hat{\vartheta}_\mathrm{g}}{\mathrm{d} t} &= \hat{\omega}_\mathrm{g} + \frac{\alpha_\mathrm{o}}{\hat{u}_\mathrm{g}}\mathrm{Im}\{ \boldsymbol{u}_\mathrm{g} \} = \omega_\mathrm{c}
+    :label: pll_polar
+
+It can be seen that the first equation in :eq:`pll_polar` is low-pass filtering of the measured PCC voltage magnitude and the second equation is the conventional angle-tracking PLL. In these coordinates, the frequency tracking :eq:`frequency_tracking` reduces to
+
+.. math::
+    \frac{\mathrm{d} \hat{\omega}_\mathrm{g}}{\mathrm{d} t} = \frac{k_\omega}{\hat{u}_\mathrm{g} } \mathrm{Im}\{ \boldsymbol{u}_\mathrm{g} \}
+    :label: frequency_tracking_polar
+
+It can be noticed that the disturbance observer with the frequency tracking equals the conventional frequency-adaptive PLL [#Kau1997]_, with the additional feature of low-pass filtering the PCC voltage magnitude. The low-pass filtered PCC voltage can be used as a feedforward term in current control [#Har2021]_.
+
+Linearized Closed-Loop System
+-----------------------------
+
+The estimation-error dynamics are analyzed by means of linearization. Using the PCC voltage as an example, the small-signal deviation about the operating point is denoted by :math:`\Delta \boldsymbol{u}_\mathrm{g} = \boldsymbol{u}_\mathrm{g} - \boldsymbol{u}_\mathrm{g0}`, where :math:`\boldsymbol{u}_\mathrm{g0}` is the operating-point quantity. From :eq:`disturbance`--:eq:`frequency_tracking`, the linearized model for the estimation-error dynamics is obtained as
+
+.. math::
+    \frac{\mathrm{d}\Delta \tilde{\boldsymbol{u}}_\mathrm{g}}{\mathrm{d}t} &= -\alpha_\mathrm{o}\Delta \tilde{\boldsymbol{u}}_\mathrm{g} + \mathrm{j}\boldsymbol{u}_\mathrm{g0} (\Delta \omega_\mathrm{g} - \Delta \hat{\omega}_\mathrm{g}) \\
+    \frac{\mathrm{d}\Delta \hat{\omega}_\mathrm{g}}{\mathrm{d}t} &= k_\omega\mathrm{Im}\left\{ \frac{\Delta \tilde{\boldsymbol{u}}_\mathrm{g}}{\boldsymbol{u}_\mathrm{g0}} \right\}
+    :label: linearized_model
+
+where :math:`\Delta \tilde{\boldsymbol{u}}_\mathrm{g} = \Delta\boldsymbol{u}_\mathrm{g} - \Delta \hat{\boldsymbol{u}}_\mathrm{g}` is the estimation error.
+
+First, assume that the grid frequency :math:`\omega_\mathrm{g}` is constant and the frequency tracking is disabled. From :eq:`linearized_model`, the closed-loop transfer function from the PCC voltage to its estimate becomes
+
+.. math::
+    \frac{\Delta\hat{\boldsymbol{u}}_\mathrm{g}(s)}{\Delta\boldsymbol{u}_\mathrm{g}(s)} = \frac{\alpha_\mathrm{o}}{s + \alpha_\mathrm{o}}
+    :label: closed_loop_pll
+
+It can be realized that both the angle and magnitude of the PCC voltage estimate converge with the bandwidth :math:`\alpha_\mathrm{o}`.
+
+Next, the frequency-tracking dynamics are also considered. From :eq:`linearized_model`, the closed-loop transfer function from the grid angular frequency to its estimate becomes
+
+.. math::
+    \frac{\Delta\hat{\omega}_\mathrm{g}(s)}{\Delta\omega_\mathrm{g}(s)}
+    = \frac{k_\omega}{s^2 + \alpha_\mathrm{o}s + k_\omega}
+    :label: closed_loop_pll_frequency_tracking
+
+Choosing :math:`k_\omega = \alpha_\mathrm{pll}^2` and :math:`\alpha_\mathrm{o} = 2\alpha_\mathrm{pll}` yields the double pole at :math:`s = -\alpha_\mathrm{pll}`, where :math:`\alpha_\mathrm{pll}` is the frequency-tracking bandwidth.
+
+This PLL in estimated PCC coordinates is implemented in the class :class:`motulator.grid.control.PLL`. The :doc:`/grid_examples/grid_following/index` examples use the PLL to synchronize with the grid.
+
+.. rubric:: References
+
+.. [#Kau1997] Kaura, Blasko, "Operation of a phase locked loop system under distorted utility conditions," IEEE Trans. Ind. Appl., 1997, https://doi.org/10.1109/28.567077
+
+.. [#Fra1997] Franklin, Powell, Workman, "Digital Control of Dynamic Systems," 3rd ed., Menlo Park, CA: Addison-Wesley, 1997
+
+.. [#Nur2024] Nurminen, Mourouvin, Hinkkanen, Kukkola, "Multifunctional grid-forming converter control based on a disturbance observer, "IEEE Trans. Power Electron., 2024, https://doi.org/10.1109/TPEL.2024.3433503
+
+.. [#Har2021] Harnefors, Kukkola, Routimo, Hinkkanen, Wang, "A universal controller for grid-connected voltage-source converters," IEEE J. Emerg. Sel. Topics Power Electron., 2021, https://doi.org/10.1109/JESTPE.2020.3039407
diff --git a/docs/_sources/drive_examples/flux_vector/index.rst b/docs/_sources/drive_examples/flux_vector/index.rst
index d5e23a13..0addef44 100644
--- a/docs/_sources/drive_examples/flux_vector/index.rst
+++ b/docs/_sources/drive_examples/flux_vector/index.rst
@@ -13,7 +13,7 @@ These examples demonstrate flux-vector control of electric machine drives [#Pel2
 
 .. [#Awa2019] Awan, Hinkkanen, Bojoi, Pellegrino, "Stator-flux-oriented control of synchronous motors: A systematic design procedure," IEEE Trans. Ind. Appl., 2019, https://doi.org/10.1109/TIA.2019.2927316
 
-.. [#Tii2024] Tiitinen, Hinkkanen, Harnefors, "Design framework for sensorless control of synchronous machine drives," IEEE Trans. Ind. Electron., 2024, https://doi.org/10.1109/TIE.2024.3429650 
+.. [#Tii2024] Tiitinen, Hinkkanen, Harnefors, "Design framework for sensorless control of synchronous machine drives," IEEE Trans. Ind. Electron., 2024, https://doi.org/10.1109/TIE.2024.3429650
 
 
 
diff --git a/docs/_sources/drive_examples/flux_vector/plot_flux_vector_im_2kw.rst b/docs/_sources/drive_examples/flux_vector/plot_flux_vector_im_2kw.rst
index 20496719..cf7f4f75 100644
--- a/docs/_sources/drive_examples/flux_vector/plot_flux_vector_im_2kw.rst
+++ b/docs/_sources/drive_examples/flux_vector/plot_flux_vector_im_2kw.rst
@@ -180,7 +180,7 @@ Plot results in per-unit values.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 9.817 seconds)
+   **Total running time of the script:** (0 minutes 10.085 seconds)
 
 
 .. _sphx_glr_download_drive_examples_flux_vector_plot_flux_vector_im_2kw.py:
diff --git a/docs/_sources/drive_examples/flux_vector/plot_flux_vector_pmsm_2kw.rst b/docs/_sources/drive_examples/flux_vector/plot_flux_vector_pmsm_2kw.rst
index ced10821..5ccada59 100644
--- a/docs/_sources/drive_examples/flux_vector/plot_flux_vector_pmsm_2kw.rst
+++ b/docs/_sources/drive_examples/flux_vector/plot_flux_vector_pmsm_2kw.rst
@@ -168,7 +168,7 @@ Plot results in per-unit values.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 5.230 seconds)
+   **Total running time of the script:** (0 minutes 5.234 seconds)
 
 
 .. _sphx_glr_download_drive_examples_flux_vector_plot_flux_vector_pmsm_2kw.py:
diff --git a/docs/_sources/drive_examples/flux_vector/plot_flux_vector_pmsyrm_5kw.rst b/docs/_sources/drive_examples/flux_vector/plot_flux_vector_pmsyrm_5kw.rst
index f6b60870..c2757c68 100644
--- a/docs/_sources/drive_examples/flux_vector/plot_flux_vector_pmsyrm_5kw.rst
+++ b/docs/_sources/drive_examples/flux_vector/plot_flux_vector_pmsyrm_5kw.rst
@@ -357,7 +357,7 @@ Plot results in per-unit values.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 15.570 seconds)
+   **Total running time of the script:** (0 minutes 15.814 seconds)
 
 
 .. _sphx_glr_download_drive_examples_flux_vector_plot_flux_vector_pmsyrm_5kw.py:
diff --git a/docs/_sources/drive_examples/flux_vector/plot_flux_vector_syrm_7kw.rst b/docs/_sources/drive_examples/flux_vector/plot_flux_vector_syrm_7kw.rst
index 8a8a712a..5c26c709 100644
--- a/docs/_sources/drive_examples/flux_vector/plot_flux_vector_syrm_7kw.rst
+++ b/docs/_sources/drive_examples/flux_vector/plot_flux_vector_syrm_7kw.rst
@@ -280,7 +280,7 @@ machine has no magnets.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 15.097 seconds)
+   **Total running time of the script:** (0 minutes 15.326 seconds)
 
 
 .. _sphx_glr_download_drive_examples_flux_vector_plot_flux_vector_syrm_7kw.py:
diff --git a/docs/_sources/drive_examples/flux_vector/sg_execution_times.rst b/docs/_sources/drive_examples/flux_vector/sg_execution_times.rst
index 26a1d16b..582723e3 100644
--- a/docs/_sources/drive_examples/flux_vector/sg_execution_times.rst
+++ b/docs/_sources/drive_examples/flux_vector/sg_execution_times.rst
@@ -6,7 +6,7 @@
 
 Computation times
 =================
-**00:45.714** total execution time for 4 files **from drive_examples/flux_vector**:
+**00:46.459** total execution time for 4 files **from drive_examples/flux_vector**:
 
 .. container::
 
@@ -33,14 +33,14 @@ Computation times
      - Time
      - Mem (MB)
    * - :ref:`sphx_glr_drive_examples_flux_vector_plot_flux_vector_pmsyrm_5kw.py` (``plot_flux_vector_pmsyrm_5kw.py``)
-     - 00:15.570
+     - 00:15.814
      - 0.0
    * - :ref:`sphx_glr_drive_examples_flux_vector_plot_flux_vector_syrm_7kw.py` (``plot_flux_vector_syrm_7kw.py``)
-     - 00:15.097
+     - 00:15.326
      - 0.0
    * - :ref:`sphx_glr_drive_examples_flux_vector_plot_flux_vector_im_2kw.py` (``plot_flux_vector_im_2kw.py``)
-     - 00:09.817
+     - 00:10.085
      - 0.0
    * - :ref:`sphx_glr_drive_examples_flux_vector_plot_flux_vector_pmsm_2kw.py` (``plot_flux_vector_pmsm_2kw.py``)
-     - 00:05.230
+     - 00:05.234
      - 0.0
diff --git a/docs/_sources/drive_examples/index.rst b/docs/_sources/drive_examples/index.rst
index 94f9cf0b..12a6b4c3 100644
--- a/docs/_sources/drive_examples/index.rst
+++ b/docs/_sources/drive_examples/index.rst
@@ -155,7 +155,7 @@ These examples shows operation of an induction machine under open-loop V/Hz cont
 
 .. raw:: html
 
-    
    
+    
    
 
 .. only:: html
 
@@ -172,7 +172,7 @@ These examples shows operation of an induction machine under open-loop V/Hz cont
 
 .. raw:: html
 
-    
    
+    
    
 
 .. only:: html
 
@@ -331,7 +331,7 @@ These examples demonstrate flux-vector control of electric machine drives [#Pel2
 
 .. [#Awa2019] Awan, Hinkkanen, Bojoi, Pellegrino, "Stator-flux-oriented control of synchronous motors: A systematic design procedure," IEEE Trans. Ind. Appl., 2019, https://doi.org/10.1109/TIA.2019.2927316
 
-.. [#Tii2024] Tiitinen, Hinkkanen, Harnefors, "Design framework for sensorless control of synchronous machine drives," IEEE Trans. Ind. Electron., 2024, https://doi.org/10.1109/TIE.2024.3429650 
+.. [#Tii2024] Tiitinen, Hinkkanen, Harnefors, "Design framework for sensorless control of synchronous machine drives," IEEE Trans. Ind. Electron., 2024, https://doi.org/10.1109/TIE.2024.3429650
 
 
 
diff --git a/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_im_2kw.rst b/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_im_2kw.rst
index 4c6142b9..2ece0856 100644
--- a/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_im_2kw.rst
+++ b/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_im_2kw.rst
@@ -197,7 +197,7 @@ the results in SI units.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 8.944 seconds)
+   **Total running time of the script:** (0 minutes 9.176 seconds)
 
 
 .. _sphx_glr_download_drive_examples_obs_vhz_plot_obs_vhz_ctrl_im_2kw.py:
diff --git a/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_pmsm_2kw.rst b/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_pmsm_2kw.rst
index 4727d990..e168e279 100644
--- a/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_pmsm_2kw.rst
+++ b/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_pmsm_2kw.rst
@@ -175,7 +175,7 @@ the results in SI units.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 24.138 seconds)
+   **Total running time of the script:** (0 minutes 24.366 seconds)
 
 
 .. _sphx_glr_download_drive_examples_obs_vhz_plot_obs_vhz_ctrl_pmsm_2kw.py:
diff --git a/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_pmsm_2kw_two_mass.rst b/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_pmsm_2kw_two_mass.rst
index 8832cdbf..4239203e 100644
--- a/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_pmsm_2kw_two_mass.rst
+++ b/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_pmsm_2kw_two_mass.rst
@@ -265,7 +265,7 @@ rotor speed w_M.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 4.729 seconds)
+   **Total running time of the script:** (0 minutes 4.735 seconds)
 
 
 .. _sphx_glr_download_drive_examples_obs_vhz_plot_obs_vhz_ctrl_pmsm_2kw_two_mass.py:
diff --git a/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_pmsyrm_thor.rst b/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_pmsyrm_thor.rst
index 796d0274..47510a21 100644
--- a/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_pmsyrm_thor.rst
+++ b/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_pmsyrm_thor.rst
@@ -287,7 +287,7 @@ the results in SI units.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 51.271 seconds)
+   **Total running time of the script:** (0 minutes 52.301 seconds)
 
 
 .. _sphx_glr_download_drive_examples_obs_vhz_plot_obs_vhz_ctrl_pmsyrm_thor.py:
diff --git a/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_syrm_7kw.rst b/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_syrm_7kw.rst
index 01c2d43b..d044c05a 100644
--- a/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_syrm_7kw.rst
+++ b/docs/_sources/drive_examples/obs_vhz/plot_obs_vhz_ctrl_syrm_7kw.rst
@@ -266,7 +266,7 @@ the results in SI units.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 28.260 seconds)
+   **Total running time of the script:** (0 minutes 28.699 seconds)
 
 
 .. _sphx_glr_download_drive_examples_obs_vhz_plot_obs_vhz_ctrl_syrm_7kw.py:
diff --git a/docs/_sources/drive_examples/obs_vhz/sg_execution_times.rst b/docs/_sources/drive_examples/obs_vhz/sg_execution_times.rst
index 5d01c1d9..6a1fe866 100644
--- a/docs/_sources/drive_examples/obs_vhz/sg_execution_times.rst
+++ b/docs/_sources/drive_examples/obs_vhz/sg_execution_times.rst
@@ -6,7 +6,7 @@
 
 Computation times
 =================
-**01:57.341** total execution time for 5 files **from drive_examples/obs_vhz**:
+**01:59.278** total execution time for 5 files **from drive_examples/obs_vhz**:
 
 .. container::
 
@@ -33,17 +33,17 @@ Computation times
      - Time
      - Mem (MB)
    * - :ref:`sphx_glr_drive_examples_obs_vhz_plot_obs_vhz_ctrl_pmsyrm_thor.py` (``plot_obs_vhz_ctrl_pmsyrm_thor.py``)
-     - 00:51.271
+     - 00:52.301
      - 0.0
    * - :ref:`sphx_glr_drive_examples_obs_vhz_plot_obs_vhz_ctrl_syrm_7kw.py` (``plot_obs_vhz_ctrl_syrm_7kw.py``)
-     - 00:28.260
+     - 00:28.699
      - 0.0
    * - :ref:`sphx_glr_drive_examples_obs_vhz_plot_obs_vhz_ctrl_pmsm_2kw.py` (``plot_obs_vhz_ctrl_pmsm_2kw.py``)
-     - 00:24.138
+     - 00:24.366
      - 0.0
    * - :ref:`sphx_glr_drive_examples_obs_vhz_plot_obs_vhz_ctrl_im_2kw.py` (``plot_obs_vhz_ctrl_im_2kw.py``)
-     - 00:08.944
+     - 00:09.176
      - 0.0
    * - :ref:`sphx_glr_drive_examples_obs_vhz_plot_obs_vhz_ctrl_pmsm_2kw_two_mass.py` (``plot_obs_vhz_ctrl_pmsm_2kw_two_mass.py``)
-     - 00:04.729
+     - 00:04.735
      - 0.0
diff --git a/docs/_sources/drive_examples/signal_inj/plot_signal_inj_pmsm_2kw.rst b/docs/_sources/drive_examples/signal_inj/plot_signal_inj_pmsm_2kw.rst
index 29c67954..eada6466 100644
--- a/docs/_sources/drive_examples/signal_inj/plot_signal_inj_pmsm_2kw.rst
+++ b/docs/_sources/drive_examples/signal_inj/plot_signal_inj_pmsm_2kw.rst
@@ -210,7 +210,7 @@ Plot results in per-unit values.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 12.540 seconds)
+   **Total running time of the script:** (0 minutes 12.848 seconds)
 
 
 .. _sphx_glr_download_drive_examples_signal_inj_plot_signal_inj_pmsm_2kw.py:
diff --git a/docs/_sources/drive_examples/signal_inj/plot_signal_inj_syrm_7kw.rst b/docs/_sources/drive_examples/signal_inj/plot_signal_inj_syrm_7kw.rst
index 457f8704..730957e2 100644
--- a/docs/_sources/drive_examples/signal_inj/plot_signal_inj_syrm_7kw.rst
+++ b/docs/_sources/drive_examples/signal_inj/plot_signal_inj_syrm_7kw.rst
@@ -211,7 +211,7 @@ Plot results in per-unit values.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 12.463 seconds)
+   **Total running time of the script:** (0 minutes 12.693 seconds)
 
 
 .. _sphx_glr_download_drive_examples_signal_inj_plot_signal_inj_syrm_7kw.py:
diff --git a/docs/_sources/drive_examples/signal_inj/sg_execution_times.rst b/docs/_sources/drive_examples/signal_inj/sg_execution_times.rst
index 436fbb2d..eb43ed0c 100644
--- a/docs/_sources/drive_examples/signal_inj/sg_execution_times.rst
+++ b/docs/_sources/drive_examples/signal_inj/sg_execution_times.rst
@@ -6,7 +6,7 @@
 
 Computation times
 =================
-**00:25.002** total execution time for 2 files **from drive_examples/signal_inj**:
+**00:25.540** total execution time for 2 files **from drive_examples/signal_inj**:
 
 .. container::
 
@@ -33,8 +33,8 @@ Computation times
      - Time
      - Mem (MB)
    * - :ref:`sphx_glr_drive_examples_signal_inj_plot_signal_inj_pmsm_2kw.py` (``plot_signal_inj_pmsm_2kw.py``)
-     - 00:12.540
+     - 00:12.848
      - 0.0
    * - :ref:`sphx_glr_drive_examples_signal_inj_plot_signal_inj_syrm_7kw.py` (``plot_signal_inj_syrm_7kw.py``)
-     - 00:12.463
+     - 00:12.693
      - 0.0
diff --git a/docs/_sources/drive_examples/vector/plot_vector_ctrl_im_2kw.rst b/docs/_sources/drive_examples/vector/plot_vector_ctrl_im_2kw.rst
index 578fa492..bc06b2e9 100644
--- a/docs/_sources/drive_examples/vector/plot_vector_ctrl_im_2kw.rst
+++ b/docs/_sources/drive_examples/vector/plot_vector_ctrl_im_2kw.rst
@@ -223,7 +223,7 @@ the results in SI units.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 4.032 seconds)
+   **Total running time of the script:** (0 minutes 4.051 seconds)
 
 
 .. _sphx_glr_download_drive_examples_vector_plot_vector_ctrl_im_2kw.py:
diff --git a/docs/_sources/drive_examples/vector/plot_vector_ctrl_im_2kw_tq_mode.rst b/docs/_sources/drive_examples/vector/plot_vector_ctrl_im_2kw_tq_mode.rst
index 7f7dcaf9..daaa702f 100644
--- a/docs/_sources/drive_examples/vector/plot_vector_ctrl_im_2kw_tq_mode.rst
+++ b/docs/_sources/drive_examples/vector/plot_vector_ctrl_im_2kw_tq_mode.rst
@@ -179,7 +179,7 @@ the results in SI units.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 4.635 seconds)
+   **Total running time of the script:** (0 minutes 4.663 seconds)
 
 
 .. _sphx_glr_download_drive_examples_vector_plot_vector_ctrl_im_2kw_tq_mode.py:
diff --git a/docs/_sources/drive_examples/vector/plot_vector_ctrl_pmsm_2kw.rst b/docs/_sources/drive_examples/vector/plot_vector_ctrl_pmsm_2kw.rst
index 4ccfc460..12dd943f 100644
--- a/docs/_sources/drive_examples/vector/plot_vector_ctrl_pmsm_2kw.rst
+++ b/docs/_sources/drive_examples/vector/plot_vector_ctrl_pmsm_2kw.rst
@@ -157,7 +157,7 @@ Create the simulation object and simulate it.
 
  .. code-block:: none
 
-    Simulation time: 4.26 s
+    Simulation time: 4.30 s
 
 
 
@@ -165,7 +165,7 @@ Create the simulation object and simulate it.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 5.296 seconds)
+   **Total running time of the script:** (0 minutes 5.359 seconds)
 
 
 .. _sphx_glr_download_drive_examples_vector_plot_vector_ctrl_pmsm_2kw.py:
diff --git a/docs/_sources/drive_examples/vector/plot_vector_ctrl_pmsm_2kw_diode.rst b/docs/_sources/drive_examples/vector/plot_vector_ctrl_pmsm_2kw_diode.rst
index 87cf84ae..94f36b0b 100644
--- a/docs/_sources/drive_examples/vector/plot_vector_ctrl_pmsm_2kw_diode.rst
+++ b/docs/_sources/drive_examples/vector/plot_vector_ctrl_pmsm_2kw_diode.rst
@@ -180,7 +180,7 @@ Create the simulation object and simulate it.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 14.156 seconds)
+   **Total running time of the script:** (0 minutes 14.408 seconds)
 
 
 .. _sphx_glr_download_drive_examples_vector_plot_vector_ctrl_pmsm_2kw_diode.py:
diff --git a/docs/_sources/drive_examples/vector/plot_vector_ctrl_pmsyrm_thor.rst b/docs/_sources/drive_examples/vector/plot_vector_ctrl_pmsyrm_thor.rst
index c82bfa02..bcab3251 100644
--- a/docs/_sources/drive_examples/vector/plot_vector_ctrl_pmsyrm_thor.rst
+++ b/docs/_sources/drive_examples/vector/plot_vector_ctrl_pmsyrm_thor.rst
@@ -207,7 +207,7 @@ Create the simulation object, simulate, and plot results in per-unit values.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 4.907 seconds)
+   **Total running time of the script:** (0 minutes 4.918 seconds)
 
 
 .. _sphx_glr_download_drive_examples_vector_plot_vector_ctrl_pmsyrm_thor.py:
diff --git a/docs/_sources/drive_examples/vector/plot_vector_ctrl_syrm_7kw.rst b/docs/_sources/drive_examples/vector/plot_vector_ctrl_syrm_7kw.rst
index b29ab65c..ee63954b 100644
--- a/docs/_sources/drive_examples/vector/plot_vector_ctrl_syrm_7kw.rst
+++ b/docs/_sources/drive_examples/vector/plot_vector_ctrl_syrm_7kw.rst
@@ -159,7 +159,7 @@ Create the simulation object, simulate, and plot results in per-unit values.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 25.594 seconds)
+   **Total running time of the script:** (0 minutes 26.194 seconds)
 
 
 .. _sphx_glr_download_drive_examples_vector_plot_vector_ctrl_syrm_7kw.py:
diff --git a/docs/_sources/drive_examples/vector/sg_execution_times.rst b/docs/_sources/drive_examples/vector/sg_execution_times.rst
index 0a84ec0a..e1bbab4b 100644
--- a/docs/_sources/drive_examples/vector/sg_execution_times.rst
+++ b/docs/_sources/drive_examples/vector/sg_execution_times.rst
@@ -6,7 +6,7 @@
 
 Computation times
 =================
-**00:58.621** total execution time for 6 files **from drive_examples/vector**:
+**00:59.592** total execution time for 6 files **from drive_examples/vector**:
 
 .. container::
 
@@ -33,20 +33,20 @@ Computation times
      - Time
      - Mem (MB)
    * - :ref:`sphx_glr_drive_examples_vector_plot_vector_ctrl_syrm_7kw.py` (``plot_vector_ctrl_syrm_7kw.py``)
-     - 00:25.594
+     - 00:26.194
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vector_plot_vector_ctrl_pmsm_2kw_diode.py` (``plot_vector_ctrl_pmsm_2kw_diode.py``)
-     - 00:14.156
+     - 00:14.408
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vector_plot_vector_ctrl_pmsm_2kw.py` (``plot_vector_ctrl_pmsm_2kw.py``)
-     - 00:05.296
+     - 00:05.359
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vector_plot_vector_ctrl_pmsyrm_thor.py` (``plot_vector_ctrl_pmsyrm_thor.py``)
-     - 00:04.907
+     - 00:04.918
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vector_plot_vector_ctrl_im_2kw_tq_mode.py` (``plot_vector_ctrl_im_2kw_tq_mode.py``)
-     - 00:04.635
+     - 00:04.663
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vector_plot_vector_ctrl_im_2kw.py` (``plot_vector_ctrl_im_2kw.py``)
-     - 00:04.032
+     - 00:04.051
      - 0.0
diff --git a/docs/_sources/drive_examples/vhz/index.rst b/docs/_sources/drive_examples/vhz/index.rst
index c1ae2076..d236501a 100644
--- a/docs/_sources/drive_examples/vhz/index.rst
+++ b/docs/_sources/drive_examples/vhz/index.rst
@@ -17,7 +17,7 @@ These examples shows operation of an induction machine under open-loop V/Hz cont
 
 .. raw:: html
 
-    
    
+    
    
 
 .. only:: html
 
@@ -34,7 +34,7 @@ These examples shows operation of an induction machine under open-loop V/Hz cont
 
 .. raw:: html
 
-    
    
+    
    
 
 .. only:: html
 
diff --git a/docs/_sources/drive_examples/vhz/plot_vhz_ctrl_6step_im_2kw.rst b/docs/_sources/drive_examples/vhz/plot_vhz_ctrl_6step_im_2kw.rst
index 9a6718b7..e86114d1 100644
--- a/docs/_sources/drive_examples/vhz/plot_vhz_ctrl_6step_im_2kw.rst
+++ b/docs/_sources/drive_examples/vhz/plot_vhz_ctrl_6step_im_2kw.rst
@@ -21,10 +21,10 @@
 2.2-kW induction motor, 6-step mode
 ===================================
 
-This example simulates V/Hz control of a 2.2-kW induction motor drive. The 
-six-step overmodulation is enabled, which increases the fundamental voltage as 
-well as the harmonics. Since the PWM is not synchronized with the stator 
-frequency, the harmonic content also depends on the ratio between the stator 
+This example simulates V/Hz control of a 2.2-kW induction motor drive. The
+six-step overmodulation is enabled, which increases the fundamental voltage as
+well as the harmonics. Since the PWM is not synchronized with the stator
+frequency, the harmonic content also depends on the ratio between the stator
 frequency and the sampling frequency.
 
 .. GENERATED FROM PYTHON SOURCE LINES 13-22
@@ -197,7 +197,7 @@ Plot results in per-unit values.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 8.841 seconds)
+   **Total running time of the script:** (0 minutes 9.044 seconds)
 
 
 .. _sphx_glr_download_drive_examples_vhz_plot_vhz_ctrl_6step_im_2kw.py:
diff --git a/docs/_sources/drive_examples/vhz/plot_vhz_ctrl_im_2kw.rst b/docs/_sources/drive_examples/vhz/plot_vhz_ctrl_im_2kw.rst
index 48d170c6..fc17b53f 100644
--- a/docs/_sources/drive_examples/vhz/plot_vhz_ctrl_im_2kw.rst
+++ b/docs/_sources/drive_examples/vhz/plot_vhz_ctrl_im_2kw.rst
@@ -22,7 +22,7 @@
 ====================================
 
 A diode bridge, stiff three-phase grid, and a DC link is modeled. The default
-parameters in this example yield open-loop V/Hz control. 
+parameters in this example yield open-loop V/Hz control.
 
 .. GENERATED FROM PYTHON SOURCE LINES 10-19
 
@@ -217,7 +217,7 @@ Plot results in per-unit values.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 17.417 seconds)
+   **Total running time of the script:** (0 minutes 17.930 seconds)
 
 
 .. _sphx_glr_download_drive_examples_vhz_plot_vhz_ctrl_im_2kw.py:
diff --git a/docs/_sources/drive_examples/vhz/plot_vhz_ctrl_im_2kw_lc.rst b/docs/_sources/drive_examples/vhz/plot_vhz_ctrl_im_2kw_lc.rst
index 631279c4..57b091c7 100644
--- a/docs/_sources/drive_examples/vhz/plot_vhz_ctrl_im_2kw_lc.rst
+++ b/docs/_sources/drive_examples/vhz/plot_vhz_ctrl_im_2kw_lc.rst
@@ -239,7 +239,7 @@ Plot additional waveforms.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 9.666 seconds)
+   **Total running time of the script:** (0 minutes 9.861 seconds)
 
 
 .. _sphx_glr_download_drive_examples_vhz_plot_vhz_ctrl_im_2kw_lc.py:
diff --git a/docs/_sources/drive_examples/vhz/sg_execution_times.rst b/docs/_sources/drive_examples/vhz/sg_execution_times.rst
index b621a4dd..005fcf3c 100644
--- a/docs/_sources/drive_examples/vhz/sg_execution_times.rst
+++ b/docs/_sources/drive_examples/vhz/sg_execution_times.rst
@@ -6,7 +6,7 @@
 
 Computation times
 =================
-**00:35.924** total execution time for 3 files **from drive_examples/vhz**:
+**00:36.835** total execution time for 3 files **from drive_examples/vhz**:
 
 .. container::
 
@@ -33,11 +33,11 @@ Computation times
      - Time
      - Mem (MB)
    * - :ref:`sphx_glr_drive_examples_vhz_plot_vhz_ctrl_im_2kw.py` (``plot_vhz_ctrl_im_2kw.py``)
-     - 00:17.417
+     - 00:17.930
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vhz_plot_vhz_ctrl_im_2kw_lc.py` (``plot_vhz_ctrl_im_2kw_lc.py``)
-     - 00:09.666
+     - 00:09.861
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vhz_plot_vhz_ctrl_6step_im_2kw.py` (``plot_vhz_ctrl_6step_im_2kw.py``)
-     - 00:08.841
+     - 00:09.044
      - 0.0
diff --git a/docs/_sources/grid_examples/grid_following/plot_gfl_10kva.rst b/docs/_sources/grid_examples/grid_following/plot_gfl_10kva.rst
index 6094f374..f0fa6e16 100644
--- a/docs/_sources/grid_examples/grid_following/plot_gfl_10kva.rst
+++ b/docs/_sources/grid_examples/grid_following/plot_gfl_10kva.rst
@@ -66,7 +66,7 @@ Compute base values based on the nominal values.
 
 Configure the system model.
 
-.. GENERATED FROM PYTHON SOURCE LINES 27-41
+.. GENERATED FROM PYTHON SOURCE LINES 27-39
 
 .. code-block:: Python
 
@@ -80,9 +80,8 @@ Configure the system model.
 
     # Create system model
     mdl = model.GridConverterSystem(converter, ac_filter, ac_source)
+    # mdl.pwm = model.CarrierComparison()  # Uncomment to enable the PWM model
 
-    # Uncomment line below to enable the PWM model
-    # mdl.pwm = model.CarrierComparison()
 
 
 
@@ -90,12 +89,11 @@ Configure the system model.
 
 
 
-
-.. GENERATED FROM PYTHON SOURCE LINES 42-43
+.. GENERATED FROM PYTHON SOURCE LINES 40-41
 
 Configure the control system.
 
-.. GENERATED FROM PYTHON SOURCE LINES 43-48
+.. GENERATED FROM PYTHON SOURCE LINES 41-46
 
 .. code-block:: Python
 
@@ -111,11 +109,11 @@ Configure the control system.
 
 
 
-.. GENERATED FROM PYTHON SOURCE LINES 49-50
+.. GENERATED FROM PYTHON SOURCE LINES 47-48
 
 Set the time-dependent reference and disturbance signals.
 
-.. GENERATED FROM PYTHON SOURCE LINES 50-60
+.. GENERATED FROM PYTHON SOURCE LINES 48-58
 
 .. code-block:: Python
 
@@ -136,11 +134,11 @@ Set the time-dependent reference and disturbance signals.
 
 
 
-.. GENERATED FROM PYTHON SOURCE LINES 61-62
+.. GENERATED FROM PYTHON SOURCE LINES 59-60
 
 Create the simulation object and simulate it.
 
-.. GENERATED FROM PYTHON SOURCE LINES 62-66
+.. GENERATED FROM PYTHON SOURCE LINES 60-64
 
 .. code-block:: Python
 
@@ -155,11 +153,11 @@ Create the simulation object and simulate it.
 
 
 
-.. GENERATED FROM PYTHON SOURCE LINES 67-68
+.. GENERATED FROM PYTHON SOURCE LINES 65-66
 
 Plot the results.
 
-.. GENERATED FROM PYTHON SOURCE LINES 68-75
+.. GENERATED FROM PYTHON SOURCE LINES 66-73
 
 .. code-block:: Python
 
@@ -197,7 +195,7 @@ Plot the results.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 1.230 seconds)
+   **Total running time of the script:** (0 minutes 1.254 seconds)
 
 
 .. _sphx_glr_download_grid_examples_grid_following_plot_gfl_10kva.py:
diff --git a/docs/_sources/grid_examples/grid_following/plot_gfl_dc_bus_10kva.rst b/docs/_sources/grid_examples/grid_following/plot_gfl_dc_bus_10kva.rst
index 81355060..46391db4 100644
--- a/docs/_sources/grid_examples/grid_following/plot_gfl_dc_bus_10kva.rst
+++ b/docs/_sources/grid_examples/grid_following/plot_gfl_dc_bus_10kva.rst
@@ -195,7 +195,7 @@ Plot the results.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 1.204 seconds)
+   **Total running time of the script:** (0 minutes 1.220 seconds)
 
 
 .. _sphx_glr_download_grid_examples_grid_following_plot_gfl_dc_bus_10kva.py:
diff --git a/docs/_sources/grid_examples/grid_following/plot_gfl_lcl_10kva.rst b/docs/_sources/grid_examples/grid_following/plot_gfl_lcl_10kva.rst
index 6342e3fd..0969d5c7 100644
--- a/docs/_sources/grid_examples/grid_following/plot_gfl_lcl_10kva.rst
+++ b/docs/_sources/grid_examples/grid_following/plot_gfl_lcl_10kva.rst
@@ -184,7 +184,7 @@ Plot the results.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 1.199 seconds)
+   **Total running time of the script:** (0 minutes 1.246 seconds)
 
 
 .. _sphx_glr_download_grid_examples_grid_following_plot_gfl_lcl_10kva.py:
diff --git a/docs/_sources/grid_examples/grid_following/sg_execution_times.rst b/docs/_sources/grid_examples/grid_following/sg_execution_times.rst
index 9ebf4c5f..debf4746 100644
--- a/docs/_sources/grid_examples/grid_following/sg_execution_times.rst
+++ b/docs/_sources/grid_examples/grid_following/sg_execution_times.rst
@@ -6,7 +6,7 @@
 
 Computation times
 =================
-**00:03.633** total execution time for 3 files **from grid_examples/grid_following**:
+**00:03.719** total execution time for 3 files **from grid_examples/grid_following**:
 
 .. container::
 
@@ -33,11 +33,11 @@ Computation times
      - Time
      - Mem (MB)
    * - :ref:`sphx_glr_grid_examples_grid_following_plot_gfl_10kva.py` (``plot_gfl_10kva.py``)
-     - 00:01.230
-     - 0.0
-   * - :ref:`sphx_glr_grid_examples_grid_following_plot_gfl_dc_bus_10kva.py` (``plot_gfl_dc_bus_10kva.py``)
-     - 00:01.204
+     - 00:01.254
      - 0.0
    * - :ref:`sphx_glr_grid_examples_grid_following_plot_gfl_lcl_10kva.py` (``plot_gfl_lcl_10kva.py``)
-     - 00:01.199
+     - 00:01.246
+     - 0.0
+   * - :ref:`sphx_glr_grid_examples_grid_following_plot_gfl_dc_bus_10kva.py` (``plot_gfl_dc_bus_10kva.py``)
+     - 00:01.220
      - 0.0
diff --git a/docs/_sources/grid_examples/grid_forming/plot_gfm_obs_13kva.rst b/docs/_sources/grid_examples/grid_forming/plot_gfm_obs_13kva.rst
index a8c3834b..f336c6e5 100644
--- a/docs/_sources/grid_examples/grid_forming/plot_gfm_obs_13kva.rst
+++ b/docs/_sources/grid_examples/grid_forming/plot_gfm_obs_13kva.rst
@@ -71,7 +71,7 @@ Configure the system model.
 
     # Filter and grid parameters
     par = ACFilterPars(L_fc=.15*base.L, R_fc=.05*base.Z, L_g=.74*base.L)
-    # par.L_g = 0 # Uncomment this line to simulate a strong grid
+    # par.L_g = 0  # Uncomment this line to simulate a strong grid
     ac_filter = model.ACFilter(par)
     ac_source = model.ThreePhaseVoltageSource(w_g=base.w, abs_e_g=base.u)
     # Inverter with constant DC voltage
@@ -103,8 +103,8 @@ Configure the control system.
         nom_u=base.u,
         nom_w=base.w,
         max_i=1.3*base.i,
-        T_s=100e-6,
-        R_a=.2*base.Z)
+        R_a=.2*base.Z,
+        T_s=100e-6)
 
     # Create the control system
     ctrl = control.ObserverBasedGridFormingControl(cfg)
@@ -202,7 +202,7 @@ Plot the results.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 8.097 seconds)
+   **Total running time of the script:** (0 minutes 8.092 seconds)
 
 
 .. _sphx_glr_download_grid_examples_grid_forming_plot_gfm_obs_13kva.py:
diff --git a/docs/_sources/grid_examples/grid_forming/plot_gfm_rfpsc_13kva.rst b/docs/_sources/grid_examples/grid_forming/plot_gfm_rfpsc_13kva.rst
index 38039d22..f0b13112 100644
--- a/docs/_sources/grid_examples/grid_forming/plot_gfm_rfpsc_13kva.rst
+++ b/docs/_sources/grid_examples/grid_forming/plot_gfm_rfpsc_13kva.rst
@@ -68,7 +68,7 @@ Configure the system model.
 
 
     # Filter and grid
-    par = ACFilterPars(L_fc=.15*base.L, L_g=.74*base.L)
+    par = ACFilterPars(L_fc=.15*base.L, R_fc=.05*base.Z, L_g=.74*base.L)
     # par.L_g = 0  # Uncomment this line to simulate a strong grid
     ac_filter = model.ACFilter(par)
     # Grid voltage source with constant frequency and voltage magnitude
@@ -90,14 +90,19 @@ Configure the system model.
 
 Configure the control system.
 
-.. GENERATED FROM PYTHON SOURCE LINES 38-46
+.. GENERATED FROM PYTHON SOURCE LINES 38-51
 
 .. code-block:: Python
 
 
     # Control configuration parameters
     cfg = control.PowerSynchronizationControlCfg(
-        nom_u=base.u, nom_w=base.w, max_i=1.3*base.i, T_s=100e-6, R_a=.2*base.Z)
+        nom_u=base.u,
+        nom_w=base.w,
+        max_i=1.3*base.i,
+        R=.05*base.Z,
+        R_a=.2*base.Z,
+        T_s=100e-6)
 
     # Create the control system
     ctrl = control.PowerSynchronizationControl(cfg)
@@ -109,11 +114,11 @@ Configure the control system.
 
 
 
-.. GENERATED FROM PYTHON SOURCE LINES 47-48
+.. GENERATED FROM PYTHON SOURCE LINES 52-53
 
 Set the references for converter output voltage magnitude and active power.
 
-.. GENERATED FROM PYTHON SOURCE LINES 48-56
+.. GENERATED FROM PYTHON SOURCE LINES 53-61
 
 .. code-block:: Python
 
@@ -132,11 +137,11 @@ Set the references for converter output voltage magnitude and active power.
 
 
 
-.. GENERATED FROM PYTHON SOURCE LINES 57-58
+.. GENERATED FROM PYTHON SOURCE LINES 62-63
 
 Create the simulation object and simulate it.
 
-.. GENERATED FROM PYTHON SOURCE LINES 58-62
+.. GENERATED FROM PYTHON SOURCE LINES 63-67
 
 .. code-block:: Python
 
@@ -151,11 +156,11 @@ Create the simulation object and simulate it.
 
 
 
-.. GENERATED FROM PYTHON SOURCE LINES 63-64
+.. GENERATED FROM PYTHON SOURCE LINES 68-69
 
 Plot the results.
 
-.. GENERATED FROM PYTHON SOURCE LINES 64-66
+.. GENERATED FROM PYTHON SOURCE LINES 69-71
 
 .. code-block:: Python
 
@@ -188,7 +193,7 @@ Plot the results.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 7.722 seconds)
+   **Total running time of the script:** (0 minutes 7.786 seconds)
 
 
 .. _sphx_glr_download_grid_examples_grid_forming_plot_gfm_rfpsc_13kva.py:
diff --git a/docs/_sources/grid_examples/grid_forming/sg_execution_times.rst b/docs/_sources/grid_examples/grid_forming/sg_execution_times.rst
index f6f3222e..0c1f34f5 100644
--- a/docs/_sources/grid_examples/grid_forming/sg_execution_times.rst
+++ b/docs/_sources/grid_examples/grid_forming/sg_execution_times.rst
@@ -6,7 +6,7 @@
 
 Computation times
 =================
-**00:15.818** total execution time for 2 files **from grid_examples/grid_forming**:
+**00:15.878** total execution time for 2 files **from grid_examples/grid_forming**:
 
 .. container::
 
@@ -33,8 +33,8 @@ Computation times
      - Time
      - Mem (MB)
    * - :ref:`sphx_glr_grid_examples_grid_forming_plot_gfm_obs_13kva.py` (``plot_gfm_obs_13kva.py``)
-     - 00:08.097
+     - 00:08.092
      - 0.0
    * - :ref:`sphx_glr_grid_examples_grid_forming_plot_gfm_rfpsc_13kva.py` (``plot_gfm_rfpsc_13kva.py``)
-     - 00:07.722
+     - 00:07.786
      - 0.0
diff --git a/docs/_sources/sg_execution_times.rst b/docs/_sources/sg_execution_times.rst
index f0029e1f..c64e11d1 100644
--- a/docs/_sources/sg_execution_times.rst
+++ b/docs/_sources/sg_execution_times.rst
@@ -6,7 +6,7 @@
 
 Computation times
 =================
-**05:02.054** total execution time for 25 files **from all galleries**:
+**05:07.301** total execution time for 25 files **from all galleries**:
 
 .. container::
 
@@ -33,77 +33,77 @@ Computation times
      - Time
      - Mem (MB)
    * - :ref:`sphx_glr_drive_examples_obs_vhz_plot_obs_vhz_ctrl_pmsyrm_thor.py` (``../../examples/drive/obs_vhz/plot_obs_vhz_ctrl_pmsyrm_thor.py``)
-     - 00:51.271
+     - 00:52.301
      - 0.0
    * - :ref:`sphx_glr_drive_examples_obs_vhz_plot_obs_vhz_ctrl_syrm_7kw.py` (``../../examples/drive/obs_vhz/plot_obs_vhz_ctrl_syrm_7kw.py``)
-     - 00:28.260
+     - 00:28.699
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vector_plot_vector_ctrl_syrm_7kw.py` (``../../examples/drive/vector/plot_vector_ctrl_syrm_7kw.py``)
-     - 00:25.594
+     - 00:26.194
      - 0.0
    * - :ref:`sphx_glr_drive_examples_obs_vhz_plot_obs_vhz_ctrl_pmsm_2kw.py` (``../../examples/drive/obs_vhz/plot_obs_vhz_ctrl_pmsm_2kw.py``)
-     - 00:24.138
+     - 00:24.366
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vhz_plot_vhz_ctrl_im_2kw.py` (``../../examples/drive/vhz/plot_vhz_ctrl_im_2kw.py``)
-     - 00:17.417
+     - 00:17.930
      - 0.0
    * - :ref:`sphx_glr_drive_examples_flux_vector_plot_flux_vector_pmsyrm_5kw.py` (``../../examples/drive/flux_vector/plot_flux_vector_pmsyrm_5kw.py``)
-     - 00:15.570
+     - 00:15.814
      - 0.0
    * - :ref:`sphx_glr_drive_examples_flux_vector_plot_flux_vector_syrm_7kw.py` (``../../examples/drive/flux_vector/plot_flux_vector_syrm_7kw.py``)
-     - 00:15.097
+     - 00:15.326
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vector_plot_vector_ctrl_pmsm_2kw_diode.py` (``../../examples/drive/vector/plot_vector_ctrl_pmsm_2kw_diode.py``)
-     - 00:14.156
+     - 00:14.408
      - 0.0
    * - :ref:`sphx_glr_drive_examples_signal_inj_plot_signal_inj_pmsm_2kw.py` (``../../examples/drive/signal_inj/plot_signal_inj_pmsm_2kw.py``)
-     - 00:12.540
+     - 00:12.848
      - 0.0
    * - :ref:`sphx_glr_drive_examples_signal_inj_plot_signal_inj_syrm_7kw.py` (``../../examples/drive/signal_inj/plot_signal_inj_syrm_7kw.py``)
-     - 00:12.463
+     - 00:12.693
      - 0.0
    * - :ref:`sphx_glr_drive_examples_flux_vector_plot_flux_vector_im_2kw.py` (``../../examples/drive/flux_vector/plot_flux_vector_im_2kw.py``)
-     - 00:09.817
+     - 00:10.085
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vhz_plot_vhz_ctrl_im_2kw_lc.py` (``../../examples/drive/vhz/plot_vhz_ctrl_im_2kw_lc.py``)
-     - 00:09.666
+     - 00:09.861
      - 0.0
    * - :ref:`sphx_glr_drive_examples_obs_vhz_plot_obs_vhz_ctrl_im_2kw.py` (``../../examples/drive/obs_vhz/plot_obs_vhz_ctrl_im_2kw.py``)
-     - 00:08.944
+     - 00:09.176
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vhz_plot_vhz_ctrl_6step_im_2kw.py` (``../../examples/drive/vhz/plot_vhz_ctrl_6step_im_2kw.py``)
-     - 00:08.841
+     - 00:09.044
      - 0.0
    * - :ref:`sphx_glr_grid_examples_grid_forming_plot_gfm_obs_13kva.py` (``../../examples/grid/grid_forming/plot_gfm_obs_13kva.py``)
-     - 00:08.097
+     - 00:08.092
      - 0.0
    * - :ref:`sphx_glr_grid_examples_grid_forming_plot_gfm_rfpsc_13kva.py` (``../../examples/grid/grid_forming/plot_gfm_rfpsc_13kva.py``)
-     - 00:07.722
+     - 00:07.786
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vector_plot_vector_ctrl_pmsm_2kw.py` (``../../examples/drive/vector/plot_vector_ctrl_pmsm_2kw.py``)
-     - 00:05.296
+     - 00:05.359
      - 0.0
    * - :ref:`sphx_glr_drive_examples_flux_vector_plot_flux_vector_pmsm_2kw.py` (``../../examples/drive/flux_vector/plot_flux_vector_pmsm_2kw.py``)
-     - 00:05.230
+     - 00:05.234
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vector_plot_vector_ctrl_pmsyrm_thor.py` (``../../examples/drive/vector/plot_vector_ctrl_pmsyrm_thor.py``)
-     - 00:04.907
+     - 00:04.918
      - 0.0
    * - :ref:`sphx_glr_drive_examples_obs_vhz_plot_obs_vhz_ctrl_pmsm_2kw_two_mass.py` (``../../examples/drive/obs_vhz/plot_obs_vhz_ctrl_pmsm_2kw_two_mass.py``)
-     - 00:04.729
+     - 00:04.735
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vector_plot_vector_ctrl_im_2kw_tq_mode.py` (``../../examples/drive/vector/plot_vector_ctrl_im_2kw_tq_mode.py``)
-     - 00:04.635
+     - 00:04.663
      - 0.0
    * - :ref:`sphx_glr_drive_examples_vector_plot_vector_ctrl_im_2kw.py` (``../../examples/drive/vector/plot_vector_ctrl_im_2kw.py``)
-     - 00:04.032
+     - 00:04.051
      - 0.0
    * - :ref:`sphx_glr_grid_examples_grid_following_plot_gfl_10kva.py` (``../../examples/grid/grid_following/plot_gfl_10kva.py``)
-     - 00:01.230
-     - 0.0
-   * - :ref:`sphx_glr_grid_examples_grid_following_plot_gfl_dc_bus_10kva.py` (``../../examples/grid/grid_following/plot_gfl_dc_bus_10kva.py``)
-     - 00:01.204
+     - 00:01.254
      - 0.0
    * - :ref:`sphx_glr_grid_examples_grid_following_plot_gfl_lcl_10kva.py` (``../../examples/grid/grid_following/plot_gfl_lcl_10kva.py``)
-     - 00:01.199
+     - 00:01.246
+     - 0.0
+   * - :ref:`sphx_glr_grid_examples_grid_following_plot_gfl_dc_bus_10kva.py` (``../../examples/grid/grid_following/plot_gfl_dc_bus_10kva.py``)
+     - 00:01.220
      - 0.0
diff --git a/docs/autoapi/motulator/common/control/index.html b/docs/autoapi/motulator/common/control/index.html
index a15c9f8b..74a80cfa 100644
--- a/docs/autoapi/motulator/common/control/index.html
+++ b/docs/autoapi/motulator/common/control/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/autoapi/motulator/common/index.html b/docs/autoapi/motulator/common/index.html
index 990ad9d1..84c0cf16 100644
--- a/docs/autoapi/motulator/common/index.html
+++ b/docs/autoapi/motulator/common/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/autoapi/motulator/common/model/index.html b/docs/autoapi/motulator/common/model/index.html
index 97f1df9b..02d55841 100644
--- a/docs/autoapi/motulator/common/model/index.html
+++ b/docs/autoapi/motulator/common/model/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/autoapi/motulator/common/utils/index.html b/docs/autoapi/motulator/common/utils/index.html
index 88b17327..29dc13e7 100644
--- a/docs/autoapi/motulator/common/utils/index.html
+++ b/docs/autoapi/motulator/common/utils/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/autoapi/motulator/drive/control/im/index.html b/docs/autoapi/motulator/drive/control/im/index.html
index d158460a..c3744c34 100644
--- a/docs/autoapi/motulator/drive/control/im/index.html
+++ b/docs/autoapi/motulator/drive/control/im/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/autoapi/motulator/drive/control/index.html b/docs/autoapi/motulator/drive/control/index.html
index 9d486a29..8c7e8d40 100644
--- a/docs/autoapi/motulator/drive/control/index.html
+++ b/docs/autoapi/motulator/drive/control/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/autoapi/motulator/drive/control/sm/index.html b/docs/autoapi/motulator/drive/control/sm/index.html
index 4163c942..81a72e50 100644
--- a/docs/autoapi/motulator/drive/control/sm/index.html
+++ b/docs/autoapi/motulator/drive/control/sm/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/autoapi/motulator/drive/index.html b/docs/autoapi/motulator/drive/index.html
index 7681848d..128894dd 100644
--- a/docs/autoapi/motulator/drive/index.html
+++ b/docs/autoapi/motulator/drive/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/autoapi/motulator/drive/model/index.html b/docs/autoapi/motulator/drive/model/index.html
index 609f0f62..2aac666a 100644
--- a/docs/autoapi/motulator/drive/model/index.html
+++ b/docs/autoapi/motulator/drive/model/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/autoapi/motulator/drive/utils/index.html b/docs/autoapi/motulator/drive/utils/index.html
index e27f2763..396e3f0b 100644
--- a/docs/autoapi/motulator/drive/utils/index.html
+++ b/docs/autoapi/motulator/drive/utils/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/autoapi/motulator/grid/control/index.html b/docs/autoapi/motulator/grid/control/index.html
index 0d1e9cca..e3fd80c7 100644
--- a/docs/autoapi/motulator/grid/control/index.html
+++ b/docs/autoapi/motulator/grid/control/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
@@ -709,8 +710,6 @@ 
    Package Contentscomplex
    Realized converter output voltage (V) in stationary
 coordinates. This signal is obtained from the PWM.
    
 
    
-
    u_gscomplex
    PCC voltage (V) in stationary coordinates.
    
-
    
 
 
    
 
    
@@ -740,7 +739,7 @@ 
    Package ContentsReturns:
 
    ref –
    
-
    Reference signals, containing the following fields:
    
+
    Reference signals, containing some of the following fields:
    
 
    
 
    
 
    u_dcfloat
    DC-bus voltage reference (V).
    
@@ -858,7 +857,7 @@ 
    Package Contents[2]. Transparent current control is also implemented.
    
 
    
 
    Parameters:
    
-
    cfg (ObserverBasedGridFormingControlCfg) – Controller configuration parameters.
    
+
    cfg (ObserverBasedGridFormingControlCfg) – Control system configuration parameters.
    
 
    
 
    
 
    Notes
    
@@ -909,9 +908,10 @@ 
    Package Contentsfloat) – Maximum current (A), peak value.
    
 
  • R (float, optional) – Total series resistance (Ω). The default is 0.
    • 
 
  • R_a (float, optional) – Active resistance (Ω). The default is 0.25*nom_u/max_i.
    • 
-
  • T_s (float, optional) – Sampling period (s). The default is 100e-6.
    • 
+
  • k_v (float, optional) – Voltage control gain. The default is alpha_o/nom_w.
    • 
 
  • alpha_c (float, optional) – Current control bandwidth (rad/s). The default is 2*pi*400.
    • 
 
  • alpha_o (float, optional) – Observer gain (rad/s). The default is 2*pi*50.
    • 
+
  • T_s (float, optional) – Sampling period (s). The default is 100e-6.
    • 
 
 
    
 
    
@@ -997,9 +997,10 @@ 
    Package Contentsfloat) – Nominal grid voltage (V), line-to-neutral peak value.
    
 
  • nom_w (float) – Nominal grid angular frequency (rad/s).
    • 
 
  • max_i (float) – Maximum current (A), peak value.
    • 
+
  • R (float, optional) – Total series resistance (Ω). The default is 0.
    • 
 
  • R_a (float, optional) – Active resistance (Ω). The default is 0.25*nom_u/max_i.
    • 
-
  • T_s (float, optional) – Sampling period (s). The default is 100e-6.
    • 
 
  • w_b (float, optional) – Low-pass filter bandwidth (rad/s). The default is 2*pi*5.
    • 
+
  • T_s (float, optional) – Sampling period (s). The default is 100e-6.
    • 
 
 
    
 
diff --git a/docs/autoapi/motulator/grid/index.html b/docs/autoapi/motulator/grid/index.html
index 19156b2b..d49ccdd5 100644
--- a/docs/autoapi/motulator/grid/index.html
+++ b/docs/autoapi/motulator/grid/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/autoapi/motulator/grid/model/index.html b/docs/autoapi/motulator/grid/model/index.html
index 4a42fc77..deeb921a 100644
--- a/docs/autoapi/motulator/grid/model/index.html
+++ b/docs/autoapi/motulator/grid/model/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/autoapi/motulator/grid/utils/index.html b/docs/autoapi/motulator/grid/utils/index.html
index e47102c7..fbfc7a16 100644
--- a/docs/autoapi/motulator/grid/utils/index.html
+++ b/docs/autoapi/motulator/grid/utils/index.html
@@ -228,7 +228,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/autoapi/motulator/index.html b/docs/autoapi/motulator/index.html
index 0b929840..f7d61512 100644
--- a/docs/autoapi/motulator/index.html
+++ b/docs/autoapi/motulator/index.html
@@ -52,7 +52,7 @@
     
     
     
-    
+    
   
   
   
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
@@ -465,12 +466,12 @@ 
    Subpackages
     
       
       
    
         
    previous
    
-        
    Disturbance-Observer-Based PLL
    
+        
    Disturbance-Observer-Based Grid-Forming Control
    
       
    
         
     Grid Converters
 
 
diff --git a/docs/control/control_system.html b/docs/control/control_system.html
index 2fce165f..5711d3b5 100644
--- a/docs/control/control_system.html
+++ b/docs/control/control_system.html
@@ -230,7 +230,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/control/drive/current_ctrl.html b/docs/control/drive/current_ctrl.html
index b1fd5910..0cc09191 100644
--- a/docs/control/drive/current_ctrl.html
+++ b/docs/control/drive/current_ctrl.html
@@ -230,7 +230,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/control/drive/index.html b/docs/control/drive/index.html
index 08e10166..33951b13 100644
--- a/docs/control/drive/index.html
+++ b/docs/control/drive/index.html
@@ -229,7 +229,8 @@
 
  • Grid Converters
 
    • 
 
diff --git a/docs/control/drive/observers.html b/docs/control/drive/observers.html
index fa30c951..1b2b8171 100644
--- a/docs/control/drive/observers.html
+++ b/docs/control/drive/observers.html
@@ -230,7 +230,8 @@
 
  • Grid Converters
 
    • 
 
@@ -470,8 +471,8 @@ 
    Machine ModelElectric Machines). In a coordinate system rotating at the angular speed \(\omega_\mathrm{s}\), the rotor flux dynamics can be expressed using the stator and rotor quantities, respectively, as
    
 
    
 (1)#\[\begin{split}\frac{\mathrm{d} \boldsymbol{\psi}_\mathrm{R}}{\mathrm{d} t} + \mathrm{j}\omega_\mathrm{s}\boldsymbol{\psi}_\mathrm{R} &= \boldsymbol{u}_\mathrm{s} - R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - L_\sigma \frac{\mathrm{d} \boldsymbol{i}_\mathrm{s}}{\mathrm{d} t} - \mathrm{j} \omega_\mathrm{s}L_\sigma\boldsymbol{i}_\mathrm{s} \\
-&= R_\mathrm{R}\boldsymbol{i}_\mathrm{s} - \left(\alpha - \mathrm{j}\omega_\mathrm{m} \right)\boldsymbol{\psi}_\mathrm{R}\end{split}\]
    
-
    where \(\omega_\mathrm{m}\) is the electrical rotor angular speed.
    
+&= \underbrace{R_\mathrm{R}\boldsymbol{i}_\mathrm{s} - \left(\alpha - \mathrm{j}\omega_\mathrm{m} \right)\boldsymbol{\psi}_\mathrm{R}}_{\boldsymbol{v}}\end{split}\]
    
+
    where \(\omega_\mathrm{m}\) is the electrical rotor angular speed and \(\alpha = R_\mathrm{R}/L_\mathrm{M}\) is the inverse rotor time constant.
    
 
 
    
 
    Observer Structure#
    
@@ -479,14 +480,14 @@ 
    Observer Structure#
    
 
    Based on (1), a reduced-order observer can be formulated in a coordinate system rotating at \(\omega_\mathrm{s}\),
    
 
    
-(2)#\[\frac{\mathrm{d} \hat{\boldsymbol{\psi}}_\mathrm{R}}{\mathrm{d} t} + \mathrm{j}\omega_\mathrm{s}\hat{\boldsymbol{\psi}}_\mathrm{R} = \boldsymbol{v} + \boldsymbol{k}_1(\hat{\boldsymbol{v}} - \boldsymbol{v}) + \boldsymbol{k}_2(\hat{\boldsymbol{v}} - \boldsymbol{v})^*\]
    
+(2)#\[\frac{\mathrm{d} \hat{\boldsymbol{\psi}}_\mathrm{R}}{\mathrm{d} t} + \mathrm{j}\omega_\mathrm{s}\hat{\boldsymbol{\psi}}_\mathrm{R} = \hat{\boldsymbol{v}}_\mathrm{s} + \boldsymbol{k}_1(\hat{\boldsymbol{v}} - \hat{\boldsymbol{v}}_\mathrm{s}) + \boldsymbol{k}_2(\hat{\boldsymbol{v}} - \hat{\boldsymbol{v}}_\mathrm{s})^*\]
    
 
    where \(\boldsymbol{k}_1\) and \(\boldsymbol{k}_2\) are complex gains, the estimates are marked with the hat, and \(^*\) marks the complex conjugate. The back-emf estimates are
    
 
    
-(3)#\[\begin{split}\boldsymbol{v} &= \boldsymbol{u}_\mathrm{s} - R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - L_\sigma \frac{\mathrm{d} \boldsymbol{i}_\mathrm{s}}{\mathrm{d} t} - \mathrm{j} \omega_\mathrm{s}L_\sigma\boldsymbol{i}_\mathrm{s} \\
-    \hat{\boldsymbol{v}} &= R_\mathrm{R}\boldsymbol{i}_\mathrm{s} - \left(\alpha - \mathrm{j}\hat{\omega}_\mathrm{m} \right)\hat{\boldsymbol{\psi}}_\mathrm{R}\end{split}\]
    
-
    In sensored drives, the rotor speed estimate is replaced with the measured speed,  \(\hat{\omega}_\mathrm{m} = \omega_\mathrm{m}\). If needed, the estimates for the stator flux linkage and the electromagnetic torque, respectively, are
    
+(3)#\[\begin{split}\hat{\boldsymbol{v}}_\mathrm{s} &= \boldsymbol{u}_\mathrm{s} - \hat R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - \hat L_\sigma \frac{\mathrm{d} \boldsymbol{i}_\mathrm{s}}{\mathrm{d} t} - \mathrm{j} \omega_\mathrm{s}\hat L_\sigma\boldsymbol{i}_\mathrm{s} \\
+    \hat{\boldsymbol{v}} &= \hat R_\mathrm{R}\boldsymbol{i}_\mathrm{s} - \left(\hat \alpha - \mathrm{j}\hat{\omega}_\mathrm{m} \right)\hat{\boldsymbol{\psi}}_\mathrm{R}\end{split}\]
    
+
    In sensored drives, the rotor speed estimate is replaced with the measured speed, \(\hat{\omega}_\mathrm{m} = \omega_\mathrm{m}\). If needed, the estimates for the stator flux linkage and the electromagnetic torque, respectively, are
    
 
    
-(4)#\[\begin{split}\hat{\boldsymbol{\psi}}_\mathrm{s} &= \hat{\boldsymbol{\psi}}_\mathrm{R} + L_\sigma \boldsymbol{i}_\mathrm{s} \\
+(4)#\[\begin{split}\hat{\boldsymbol{\psi}}_\mathrm{s} &= \hat{\boldsymbol{\psi}}_\mathrm{R} + \hat L_\sigma \boldsymbol{i}_\mathrm{s} \\
 \hat{\tau}_\mathrm{M} &= \frac{3n_\mathrm{p}}{2}\mathrm{Im}\left\{\boldsymbol{i}_\mathrm{s} \hat{\boldsymbol{\psi}}_\mathrm{R}^* \right\}\end{split}\]
    
 
    It is also worth noticing that the derivative of the stator current in (3) is integrated, i.e., the noise is not amplified.
    
 
    
@@ -499,20 +500,20 @@ 
    Estimated Flux Coordinates\(\hat{\boldsymbol{\psi}}_\mathrm{R}^\mathrm{s} = \hat{\psi}_\mathrm{R}\mathrm{e}^{\mathrm{j}\hat{\vartheta}_\mathrm{s}}\), where \(\hat{\psi}_\mathrm{R}\) is the magnitude and \(\hat{\vartheta}_\mathrm{s}\) is the angle of the estimate. The observer is convenient to implement in estimated rotor flux coordinates, in which the rotor flux estimate is real, \(\hat{\boldsymbol{\psi}}_\mathrm{R} = \hat{\psi}_\mathrm{R} + \mathrm{j}0\) [2]. Under these conditions, the angular speed of the coordinates system can be solved from (2) as
    
 
    
 (5)#\[\frac{\mathrm{d}\hat{\vartheta}_\mathrm{s}}{\mathrm{d} t} = \omega_\mathrm{s}
-= \frac{\mathrm{Im} \{ \boldsymbol{v}' + \boldsymbol{k}_1(\hat{\boldsymbol{v}} - \boldsymbol{v}') + \boldsymbol{k}_2(\hat{\boldsymbol{v}} - \boldsymbol{v}')^* \} }{\hat{\psi}_\mathrm{R} + L_\sigma \mathrm{Re}\{(1 - \boldsymbol{k}_1)\boldsymbol{i}_\mathrm{s} + \boldsymbol{k}_2 \boldsymbol{i}_\mathrm{s}^* \}}\]
    
+= \frac{\mathrm{Im} \{ \hat{\boldsymbol{v}}_\mathrm{s}' + \boldsymbol{k}_1(\hat{\boldsymbol{v}} - \hat{\boldsymbol{v}}_\mathrm{s}') + \boldsymbol{k}_2(\hat{\boldsymbol{v}} - \hat{\boldsymbol{v}}_\mathrm{s}')^* \} }{\hat{\psi}_\mathrm{R} + \hat L_\sigma \mathrm{Re}\{(1 - \boldsymbol{k}_1)\boldsymbol{i}_\mathrm{s} + \boldsymbol{k}_2 \boldsymbol{i}_\mathrm{s}^*\}}\]
    
 
    where
    
 
    
-(6)#\[\boldsymbol{v}' = \boldsymbol{u}_\mathrm{s} - R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - L_\sigma \frac{\mathrm{d} \boldsymbol{i}_\mathrm{s}}{\mathrm{d} t}\]
    
+(6)#\[\hat{\boldsymbol{v}}_\mathrm{s}' = \boldsymbol{u}_\mathrm{s} - \hat R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - \hat L_\sigma \frac{\mathrm{d} \boldsymbol{i}_\mathrm{s}}{\mathrm{d} t}\]
    
 
    The flux magnitude dynamics are
    
 
    
 (7)#\[\frac{\mathrm{d} \hat{\psi}_\mathrm{R}}{\mathrm{d} t}
-= \mathrm{Re}\{ \boldsymbol{v} + \boldsymbol{k}_1(\hat{\boldsymbol{v}} - \boldsymbol{v}) + \boldsymbol{k}_2(\hat{\boldsymbol{v}} - \boldsymbol{v})^* \}\]
    
-
    Notice that the right-hand side of (5) is independent of \(\omega_\mathrm{s}\). Futhermore, in these coordinates, the condition (12) for an inherently sensorless observer reduces to \(\boldsymbol{k}_2 = \boldsymbol{k}_1\). This observer structure is implemented in the motulator.drive.control.im.Observer, where simple forward-Euler discretization is used.
    
+= \mathrm{Re}\{ \hat{\boldsymbol{v}}_\mathrm{s} + \boldsymbol{k}_1(\hat{\boldsymbol{v}} - \hat{\boldsymbol{v}}_\mathrm{s}) + \boldsymbol{k}_2(\hat{\boldsymbol{v}} - \hat{\boldsymbol{v}}_\mathrm{s})^* \}\]
    
+
    Notice that the right-hand side of (5) is independent of \(\omega_\mathrm{s}\). Furthermore, in these coordinates, the condition (12) for an inherently sensorless observer reduces to \(\boldsymbol{k}_2 = \boldsymbol{k}_1\). This observer structure is implemented in the motulator.drive.control.im.Observer, where simple forward-Euler discretization is used.
    
 
 
 
    
 
    Gain Selection#
    
-
    The estimation-error dynamics are obtained by subtracting (2) from (1). The resulting system can be linearized for analysis and gain selection purposes. Using the rotor speed as an example, the small-signal deviation about the operating point is \(\Delta \omega_\mathrm{m} = \omega_\mathrm{m} - \omega_\mathrm{m0}\), where the subscript 0 refers to the operating point. Linearization of the estimation-error dynamics leads to [3]
    
+
    The estimation-error dynamics are obtained by subtracting (2) from (1). The resulting system can be linearized for analysis and gain selection purposes. Accurate parameter estimates are assumed in the analysis. Using the rotor speed as an example, the small-signal deviation about the operating point is \(\Delta \omega_\mathrm{m} = \omega_\mathrm{m} - \omega_\mathrm{m0}\), where the subscript 0 refers to the operating point. Linearization of the estimation-error dynamics leads to [3]
    
 
    
 (8)#\[\frac{\mathrm{d} \Delta\tilde{\boldsymbol{\psi}}_\mathrm{R}}{\mathrm{d} t} =  \boldsymbol{k}_1\Delta \tilde{\boldsymbol{v}} + \boldsymbol{k}_2\Delta \tilde{\boldsymbol{v}}^* - \mathrm{j}\omega_\mathrm{s0}\Delta\tilde{\boldsymbol{\psi}}_\mathrm{R}\]
    
 
    where the estimation error of the rotor flux is \(\Delta\tilde{\boldsymbol{\psi}}_\mathrm{R} = \Delta\boldsymbol{\psi}_\mathrm{R} - \Delta\hat{\boldsymbol{\psi}}_\mathrm{R}\) and other estimation errors are marked similarly. Furthermore, the back-emf estimation error is
    
@@ -525,7 +526,7 @@ 
    Sensored Case(10)#\[\frac{\mathrm{d} \Delta\tilde{\boldsymbol{\psi}}_\mathrm{R}}{\mathrm{d} t} =  -\left[\boldsymbol{k}_1\left(\alpha - \mathrm{j}\omega_\mathrm{m0} \right) + \mathrm{j}\omega_\mathrm{s0}\right]\Delta\tilde{\boldsymbol{\psi}}_\mathrm{R}\]
    
 
    It can be noticed that the closed-loop pole could be arbitrarily placed via the gain \(\boldsymbol{k}_1\). Well-damped estimation-error dynamics can be obtained, e.g., by choosing
    
 
    
-(11)#\[\boldsymbol{k}_1 = 1 + \frac{g |\omega_\mathrm{m}|}{\alpha - \mathrm{j}\omega_\mathrm{m}}\]
    
+(11)#\[\boldsymbol{k}_1 = 1 + \frac{g |\omega_\mathrm{m}|}{\hat \alpha - \mathrm{j}\omega_\mathrm{m}}\]
    
 
    where \(g\) is a unitless positive design parameter. The corresponding pole is located at \(s = -\alpha - g |\omega_\mathrm{m0}| - \mathrm{j}\omega_\mathrm{r0}\), where \(\omega_\mathrm{r0} = \omega_\mathrm{s0} - \omega_\mathrm{m0}\) is the operating-point slip angular frequency.
    
 
    
 
    Note
    
@@ -543,8 +544,8 @@ 
    Sensorless Case\(\boldsymbol{k}_1 = k_\mathrm{d} + \mathrm{j}k_\mathrm{q}\). It can be seen that the dynamics of the rotor speed are decoupled from the flux-estimation error dynamics. The decay rate \(\sigma\) be assigned by choosing
    
 
    
-(14)#\[\boldsymbol{k}_1 = \frac{\sigma}{\alpha - \mathrm{j}\hat\omega_\mathrm{m}}\]
    
-
    which results in the characteristic polynomial \(D(s)=s^2 + 2\sigma s + \omega_\mathrm{s0}^2\). The decay rate can be selected as \(\sigma = \alpha/2 + \zeta_\infty|\hat{\omega}_\mathrm{m}|\), where \(\zeta_\infty\) is the desired damping ratio at high speed. At zero stator frequency \(\omega_\mathrm{s0} = 0\), the poles are located at \(s = 0\) and \(s = -\alpha\), which allows stable magnetizing and starting the machine.
    
+(14)#\[\boldsymbol{k}_1 = \frac{\sigma}{\hat \alpha - \mathrm{j}\hat\omega_\mathrm{m}}\]
    
+
    which results in the characteristic polynomial \(D(s)=s^2 + 2\sigma s + \omega_\mathrm{s0}^2\). The decay rate can be selected as \(\sigma = \hat \alpha/2 + \zeta_\infty|\hat{\omega}_\mathrm{m}|\), where \(\zeta_\infty\) is the desired damping ratio at high speeds. At zero stator frequency \(\omega_\mathrm{s0} = 0\), the poles are located at \(s = 0\) and \(s = -\alpha\), which allows stable magnetizing and starting the machine.
    
 
    References

    where \(\hat{\boldsymbol{\psi}}_\mathrm{s}\) is the stator flux estimate. The flux linkage is estimated by

    -(18)#\[\frac{\mathrm{d} \hat{\boldsymbol{\psi}}_\mathrm{s}}{\mathrm{d} t} = \boldsymbol{u}_\mathrm{s} - R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - \mathrm{j}\omega_\mathrm{s}\hat{\boldsymbol{\psi}}_\mathrm{s} + \boldsymbol{k}_1 \boldsymbol{e} + \boldsymbol{k}_2 \boldsymbol{e}^*\]
    +(18)#\[\frac{\mathrm{d} \hat{\boldsymbol{\psi}}_\mathrm{s}}{\mathrm{d} t} = \boldsymbol{u}_\mathrm{s} - \hat R_\mathrm{s}\boldsymbol{i}_\mathrm{s} - \mathrm{j}\omega_\mathrm{s}\hat{\boldsymbol{\psi}}_\mathrm{s} + \boldsymbol{k}_1 \boldsymbol{e} + \boldsymbol{k}_2 \boldsymbol{e}^*\]

    where \(\boldsymbol{k}_1\) and \(\boldsymbol{k}_2\) are gains (complex in a general case), the estimates are marked with the hat, and \(^*\) marks the complex conjugate.

    In the sensored mode, \(\omega_\mathrm{s} = \omega_\mathrm{m}\) is used. In the sensorless mode, the speed-adaptive structure (which would correspond to the phase-locked loop if the observer were implemented in stator coordinates) can be used to estimate the rotor angle and speed, respectively, as

    @@ -621,7 +622,7 @@

    Sensorless Case(22)#\[\hat{\boldsymbol{\psi}}_\mathrm{a} = \psi_\mathrm{f} + (L_\mathrm{d} - L_\mathrm{q}) \boldsymbol{i}_\mathrm{s}^*\]

    allows to decouple the flux-estimation error dynamics from the rotor-position dynamics. By default, the decay rate is scheduled as

    -(23)#\[\sigma = \frac{R_\mathrm{s}}{4}\left(\frac{1}{L_\mathrm{d}} + \frac{1}{L_\mathrm{q}}\right) + \zeta_\infty |\hat{\omega}_\mathrm{m} |\]
    +(23)#\[\sigma = \frac{\hat R_\mathrm{s}}{4}\left(\frac{1}{\hat L_\mathrm{d}} + \frac{1}{\hat L_\mathrm{q}}\right) + \zeta_\infty |\hat{\omega}_\mathrm{m} |\]

    where \(\zeta_\infty\) is the desired damping ratio at high speed. At zero speed, (23) places one pole at \(s = 0\) and another at \(s = -(R_\mathrm{s}/2)(1/L_\mathrm{d} + 1/L_\mathrm{q})\).

    The gains of the speed adaptation in (19) are selected as

    diff --git a/docs/control/drive/speed_ctrl.html b/docs/control/drive/speed_ctrl.html index ff2fd860..bd15a46c 100644 --- a/docs/control/drive/speed_ctrl.html +++ b/docs/control/drive/speed_ctrl.html @@ -230,7 +230,8 @@
  • Grid Converters
    • diff --git a/docs/control/grid/current_ctrl.html b/docs/control/grid/current_ctrl.html index e1a94559..fe621109 100644 --- a/docs/control/grid/current_ctrl.html +++ b/docs/control/grid/current_ctrl.html @@ -52,7 +52,7 @@ - + @@ -230,7 +230,8 @@
  • Grid Converters
    • @@ -424,11 +425,9 @@

    Contents

    @@ -454,8 +453,9 @@

    2DOF PI Controller(1)#\[\begin{split}\frac{\mathrm{d} \boldsymbol{u}_\mathrm{i}}{\mathrm{d} t} &= (\boldsymbol{k}_\mathrm{i} + \mathrm{j}\omega_\mathrm{c}\boldsymbol{k}_\mathrm{t} )\left(\boldsymbol{i}_\mathrm{c,ref} - \boldsymbol{i}_\mathrm{c}\right) \\ \boldsymbol{u}_\mathrm{c,ref} &= \boldsymbol{k}_\mathrm{t}\boldsymbol{i}_\mathrm{c,ref} - \boldsymbol{k}_\mathrm{p}\boldsymbol{i}_\mathrm{c} + \boldsymbol{u}_\mathrm{i}\end{split}\]

    where \(\boldsymbol{u}_\mathrm{c,ref}\) is the output of the controller, i.e., the converter voltage reference, \(\boldsymbol{i}_\mathrm{c}\) is the measured converter current, \(\boldsymbol{i}_\mathrm{c,ref}\) is the converter current reference, \(\boldsymbol{u}_\mathrm{i}\) is the the integral state, and \(\omega_\mathrm{c}\) is the angular speed of the coordinate system. Furthermore, \(\boldsymbol{k}_\mathrm{t}\) is the reference-feedforward gain, \(\boldsymbol{k}_\mathrm{p}\) is the proportional gain, and \(\boldsymbol{k}_\mathrm{i}\) is the integral gain.

    +
    -

    Closed-Loop System#

    +

    Closed-Loop System#

    Consider the grid model in synchronous coordinates

    (2)#\[L\frac{\mathrm{d}\boldsymbol{i}_\mathrm{c}}{\mathrm{d} t} = \boldsymbol{u}_\mathrm{c} - \boldsymbol{u}_\mathrm{g} - \mathrm{j} \omega_\mathrm{c} L \boldsymbol{i}_\mathrm{c}\]
    @@ -471,7 +471,7 @@

    Closed-Loop System\(\boldsymbol{k}_\mathrm{t}\).

    -

    Gain Selection#

    +

    Gain Selection#

    Consider the gains

    \[\boldsymbol{k}_\mathrm{p} = 2\alpha_\mathrm{c} \hat L \qquad\qquad @@ -507,7 +507,6 @@

    Gain Selection

    next

    -

    Disturbance-Observer-Based PLL

    +

    Phase-Locked Loop

     @@ -554,11 +553,9 @@

    Gain Selection

    diff --git a/docs/control/grid/dc_voltage_ctrl.html b/docs/control/grid/dc_voltage_ctrl.html index 7953cef0..f402e822 100644 --- a/docs/control/grid/dc_voltage_ctrl.html +++ b/docs/control/grid/dc_voltage_ctrl.html @@ -230,7 +230,8 @@
  • Grid Converters
    • diff --git a/docs/control/grid/grid-forming.html b/docs/control/grid/grid-forming.html new file mode 100644 index 00000000..0971af15 --- /dev/null +++ b/docs/control/grid/grid-forming.html @@ -0,0 +1,630 @@ + + + + + + + + + + + Disturbance-Observer-Based Grid-Forming Control — motulator 0.4.0 documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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    Disturbance-Observer-Based Grid-Forming Control

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    + + + + +
    + +
    +

    Disturbance-Observer-Based Grid-Forming Control#

    +

    In these notes, disturbance-observer-based grid-forming control is discussed [1]. This control method is available in the motulator.grid.control.ObserverBasedGridFormingControl class. As can be realized by comparing the examples 12.5-kVA converter, RFPSC and 12.5-kVA converter, disturbance observer, the observer-based grid-forming control can be configured to behave practically identically with reference-feedforward power-synchronization control (PSC) [2]. Compared to reference-feedforward PSC, disturbance-observer-based grid-forming control is easier to analyze and extend with different operating modes (including grid-following and transparent current-control modes). The order of these two control methods is the same and their computational complexity is comparable.

    +
    +

    System Model#

    +

    First, the system is modeled in general coordinates, whose angle with respect to the stationary coordinates is \(\vartheta_\mathrm{c}\) and the angular speed is \(\omega_\mathrm{c} = \mathrm{d}\vartheta_\mathrm{c}/\mathrm{d} t\). The dynamics of the inductor current \(\boldsymbol{i}_\mathrm{c}\) and the grid voltage \(\boldsymbol{u}_\mathrm{g}\) are modeled as

    +
    +(1)#\[\begin{split}L\frac{\mathrm{d} \boldsymbol{i}_\mathrm{c}}{\mathrm{d} t} &= \boldsymbol{u}_\mathrm{c} - \boldsymbol{u}_\mathrm{g} - \mathrm{j}\omega_\mathrm{c} L \boldsymbol{i}_\mathrm{c} \\ +\frac{\mathrm{d}\boldsymbol{u}_\mathrm{g}}{\mathrm{d} t} &= \mathrm{j}(\omega_\mathrm{g} - \omega_\mathrm{c})\boldsymbol{u}_\mathrm{g}\end{split}\]
    +

    where \(L\) is the inductance and \(\omega_\mathrm{g}\) is the grid angular frequency. The grid voltage \(\boldsymbol{u}_\mathrm{g}\) is modeled as a disturbance; this same disturbance model for the grid voltage is also used in the development of Phase-Locked Loop. The active power fed to the grid is nonlinear in the state variables

    +
    +(2)#\[p_\mathrm{g} = \frac{3}{2}\mathrm{Re}\{\boldsymbol{u}_\mathrm{g}\boldsymbol{i}_\mathrm{c}^*\}\]
    +

    For the purpose of grid-forming control, we may define the quasi-static converter voltage as an output variable as a function of the state variables,

    +
    +(3)#\[\boldsymbol{v}_\mathrm{c} = \boldsymbol{u}_\mathrm{g} + \mathrm{j}\omega_\mathrm{g} L \boldsymbol{i}_\mathrm{c}\]
    +

    This voltage is controllable (unlike the grid voltage which is the disturbance) and it equals the converter output voltage \(\boldsymbol{u}_\mathrm{c}\) in the steady state. The disturbance-observer-based PSC regulates the voltage magnitude \(|\boldsymbol{v}_\mathrm{c}|\) and the power \(p_\mathrm{g}\) fed to grid.

    +
    +
    +

    Control System#

    +

    The control system consists of a disturbance observer and a control law. The disturbance observer estimates the quasi-static converter voltage and the power fed to the grid. The control law generates the converter voltage reference based on the estimated grid voltage and power. The disturbance observer also provides the integral action for the control law. This control structure allows seamless switching between grid-forming and grid-following modes [1]. For simplicity, only the grid-forming mode is considered in the following.

    +
    +

    Disturbance Observer#

    +

    Based on (1)–(3), a disturbance observer for the grid voltage can be formed [1], [3], [4]

    +
    +(4)#\[\begin{split}\frac{\mathrm{d} \hat{\boldsymbol{u}}_\mathrm{g}}{\mathrm{d} t} &= \mathrm{j} (\hat{\omega}_\mathrm{g} - \omega_\mathrm{c})\hat{\boldsymbol{u}}_\mathrm{g} + \alpha_\mathrm{o}\left(\boldsymbol{u}_\mathrm{c} - \hat L \frac{\mathrm{d} \boldsymbol{i}_\mathrm{c}}{\mathrm{d} t} - \mathrm{j} \omega_\mathrm{c} \hat L \boldsymbol{i}_\mathrm{c} - \hat{\boldsymbol{u}}_\mathrm{g} \right) \\ +\hat{\boldsymbol{v}}_\mathrm{c} &= \hat{\boldsymbol{u}}_\mathrm{g} + \mathrm{j}\hat{\omega}_\mathrm{g} \hat L \boldsymbol{i}_\mathrm{c} \\ +\hat p_\mathrm{g} &= \frac{3}{2}\mathrm{Re}\{\hat{\boldsymbol{v}}_\mathrm{c}\boldsymbol{i}_\mathrm{c}^*\}\end{split}\]
    +

    where \(\hat \omega_\mathrm{g}\) is the nominal grid angular frequency and estimates are marked with hat.

    +
    +

    Note

    +

    A conventional phase-locked loop (PLL) can be expressed in the same disturbance observer framework, see the Phase-Locked Loop notes. It can be realized that the measured grid voltage \(\boldsymbol{u}_\mathrm{g}\) used in the conventional PLL is replaced by its converter-voltage-based estimate \(\boldsymbol{u}_\mathrm{c} - \hat L (\mathrm{d} \boldsymbol{i}_\mathrm{c}/\mathrm{d} t) - \mathrm{j} \omega_\mathrm{c} \hat L \boldsymbol{i}_\mathrm{c}\) in the disturbance observer (4).

    +
    +
    +
    +

    Control Law#

    +

    A nonlinear state feedback law is used

    +
    +(5)#\[\boldsymbol{u}_\mathrm{c,ref} = \hat{\boldsymbol{v}}_\mathrm{c} + \boldsymbol{k}_\mathrm{p} (p_\mathrm{g,ref} - \hat p_\mathrm{g}) + \boldsymbol{k}_\mathrm{v} (v_\mathrm{c,ref} - |\hat{\boldsymbol{v}}_\mathrm{c}|)\]
    +

    where \(p_\mathrm{g,ref}\) is the active power reference, \(v_\mathrm{c,ref}\) is the converter voltage magnitude reference, and \(\boldsymbol{k}_\mathrm{p}\) and \(\boldsymbol{k}_\mathrm{v}\) are the complex gains for the active-power and converter-voltage-magnitude channels, respectively. To ensure robust operation, similar to the reference-feedforward PSC, the complex gains can be selected as

    +
    +(6)#\[\boldsymbol{k}_\mathrm{p} = \frac{R_\mathrm{a}}{v_\mathrm{c,ref}} \frac{\hat{\boldsymbol{v}}_\mathrm{c}}{|\hat{\boldsymbol{v}}_\mathrm{c}|} \qquad +\boldsymbol{k}_\mathrm{v} = (1 - \mathrm{j} k_\mathrm{v}) \frac{\hat{\boldsymbol{v}}_\mathrm{c}}{|\hat{\boldsymbol{v}}_\mathrm{c}|}\]
    +

    where the gains \(R_\mathrm{a} = 0.2\) p.u. and \(k_\mathrm{v} = \alpha_\mathrm{o}/\omega_\mathrm{g}\) can be used.

    +
    +
    +

    Implementation Aspects#

    +

    To avoid the derivate on the right-hand side of (4), a new state variable \(\hat{\boldsymbol{u}}_\mathrm{g}' = \hat{\boldsymbol{u}}_\mathrm{g} + \alpha_\mathrm{o} \hat L \boldsymbol{i}_\mathrm{c}\) can be introduced [4]. Furthermore, the coordinate system for the implementation can be chosen freely. The simplest choice is to use the nominal grid frequency as the coordinate system frequency, \(\omega_\mathrm{c} = \hat \omega_\mathrm{g}\). Using these design choices, the whole control system consisting of the disturbance observer (4) and the control law (5) in the state-space form reduces to [1]

    +
    +(7)#\[\begin{split}\frac{\mathrm{d} \hat{\boldsymbol{u}}'_\mathrm{g}}{\mathrm{d} t} &= \alpha_\mathrm{o} (\boldsymbol{u}_\mathrm{c,ref} - \hat{\boldsymbol{v}}_\mathrm{c} ) \\ +\hat{\boldsymbol{v}}_\mathrm{c} &= \hat{\boldsymbol{u}}_\mathrm{g}' - (\alpha_\mathrm{o} - \mathrm{j}\hat{\omega}_\mathrm{g}) \hat L \boldsymbol{i}_\mathrm{c} \\ +\hat p_\mathrm{g} &= \frac{3}{2}\mathrm{Re}\{\hat{\boldsymbol{v}}_\mathrm{c}\boldsymbol{i}_\mathrm{c}^*\} \\ +\boldsymbol{u}_\mathrm{c,ref} &= \hat{\boldsymbol{v}}_\mathrm{c} + \boldsymbol{k}_\mathrm{p} (p_\mathrm{g,ref} - \hat p_\mathrm{g}) + \boldsymbol{k}_\mathrm{v} (v_\mathrm{c,ref} - |\hat{\boldsymbol{v}}_\mathrm{c}|)\end{split}\]
    +

    where the gains can be selected according to (6) and the converter voltage appearing in the observer has been replaced with its reference. Various control modes could be easily incorporated into the control system (7), simply by changing the feedback correction terms of the control law [1]. The switching between the modes is seamless since the control law does not have memory, but the integral action is provided by the disturbance observer (in addition to synchronization).

    +

    The control system implemented in the motulator.grid.control.ObserverBasedGridFormingControl class corresponds to (7). In the example implementation, a transparent current-control mode is implemented. In the grid-forming mode, the observer bandwidth \(\alpha_\mathrm{o} = 1\) p.u. can be used. Furthermore, the inductance estimate can be set close to the lowest expected inductance value, e.g., \(\hat L = 0.15\) p.u. Using this configuration, the robust performance from strong grids to very weak grids can be achieved. This grid-forming control method can also be used with LCL filters, similarly to reference-feedforward PSC.

    +

    References

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  • Grid Converters
    • @@ -432,7 +433,8 @@

    Grid ConvertersDC-Bus Voltage Control
  • Current Control
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  • Phase-Locked Loop
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  • Grid-Forming Control

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    Phase-Locked Loop

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    Phase-Locked Loop#

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    A phase-locked loop (PLL) is commonly used in grid-following converters to synchronize the converter output with the grid [1]. Here, we represent the PLL using the disturbance observer structure [2], which may be simpler to extend than the classical PLL. Furthermore, this structure allows to see links to synchronization methods used in grid-forming converters, see [3].

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    Disturbance Model#

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    Consider the positive-sequence voltage at the point of common coupling (PCC) in general coordinates, rotating at the angular speed \(\omega_\mathrm{c}\). The dynamics of the PCC voltage \(\boldsymbol{u}_\mathrm{g}\) can be modeled using a disturbance model as

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    +(1)#\[\frac{\mathrm{d}\boldsymbol{u}_\mathrm{g}}{\mathrm{d}t} = \mathrm{j}(\omega_\mathrm{g} - \omega_\mathrm{c})\boldsymbol{u}_\mathrm{g}\]
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    where \(\omega_\mathrm{g}\) is the grid angular frequency. If needed, this disturbance model could be extended, e.g., with a negative-sequence component, allowing to design the PLL for unbalanced grids.

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    Note

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    The disturbance model in (1) can be equivalently expressed in polar form as

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    +\[\begin{split}\frac{\mathrm{d} u_\mathrm{g}}{\mathrm{d} t} &= 0 \\ +\frac{\mathrm{d} \delta_\mathrm{g}}{\mathrm{d} t} &= \omega_\mathrm{g} - \omega_\mathrm{c} \\ +\boldsymbol{u}_\mathrm{g} &= u_\mathrm{g} \mathrm{e}^{\mathrm{j}\delta_\mathrm{g}}\end{split}\]
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    where \(\delta_\mathrm{g} = \vartheta_\mathrm{g} - \vartheta_\mathrm{c}\) is the angle of the grid voltage in general coordinates and the last equation is the output equation.

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    PLL in General Coordinates#

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    Based on (1), the disturbance observer containing the regular PLL functionality can be formulated as

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    +(2)#\[\frac{\mathrm{d}\hat{\boldsymbol{u}}_\mathrm{g}}{\mathrm{d}t} = \mathrm{j}(\hat{\omega}_\mathrm{g} - \omega_\mathrm{c})\hat{\boldsymbol{u}}_\mathrm{g} + \alpha_\mathrm{o} (\boldsymbol{u}_\mathrm{g} - \hat{\boldsymbol{u}}_\mathrm{g} )\]
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    where \(\hat{\boldsymbol{u}}_\mathrm{g}\) is the estimated PCC voltage, \(\hat{\omega}_\mathrm{g}\) is the grid angular frequency estimate (either constant corresponding to the nominal value or dynamic from grid-frequency tracking), and \(\alpha_\mathrm{o}\) is the bandwidth. If needed, the disturbance observer can be extended with the frequency tracking as

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    +(3)#\[\frac{\mathrm{d}\hat{\omega}_\mathrm{g}}{\mathrm{d}t} = k_\omega\mathrm{Im}\left\{ \frac{\boldsymbol{u}_\mathrm{g} - \hat{\boldsymbol{u}}_\mathrm{g}}{\hat{\boldsymbol{u}}_\mathrm{g}} \right\}\]
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    where \(k_\omega\) is the frequency-tracking gain. Notice that (2) and (3) are both driven by the estimation error \(\boldsymbol{u}_\mathrm{g} - \hat{\boldsymbol{u}}_\mathrm{g}\) of the PCC voltage.

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    PLL in Estimated PCC Voltage Coordinates#

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    The disturbance observer (2) can be equivalently expressed in estimated PCC voltage coordinates, where \(\hat{\boldsymbol{u}}_\mathrm{g} = \hat{u}_\mathrm{g}\) and \(\omega_\mathrm{c} = \mathrm{d} \hat{\vartheta}_\mathrm{g}/ \mathrm{d} t\), resulting in

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    +(4)#\[\begin{split}\frac{\mathrm{d}\hat{u}_\mathrm{g}}{\mathrm{d}t} &= \alpha_\mathrm{o} \left(\mathrm{Re}\{\boldsymbol{u}_\mathrm{g}\} - \hat{u}_\mathrm{g} \right) \\ +\frac{\mathrm{d} \hat{\vartheta}_\mathrm{g}}{\mathrm{d} t} &= \hat{\omega}_\mathrm{g} + \frac{\alpha_\mathrm{o}}{\hat{u}_\mathrm{g}}\mathrm{Im}\{ \boldsymbol{u}_\mathrm{g} \} = \omega_\mathrm{c}\end{split}\]
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    It can be seen that the first equation in (4) is low-pass filtering of the measured PCC voltage magnitude and the second equation is the conventional angle-tracking PLL. In these coordinates, the frequency tracking (3) reduces to

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    +(5)#\[\frac{\mathrm{d} \hat{\omega}_\mathrm{g}}{\mathrm{d} t} = \frac{k_\omega}{\hat{u}_\mathrm{g} } \mathrm{Im}\{ \boldsymbol{u}_\mathrm{g} \}\]
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    It can be noticed that the disturbance observer with the frequency tracking equals the conventional frequency-adaptive PLL [1], with the additional feature of low-pass filtering the PCC voltage magnitude. The low-pass filtered PCC voltage can be used as a feedforward term in current control [4].

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    Linearized Closed-Loop System#

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    The estimation-error dynamics are analyzed by means of linearization. Using the PCC voltage as an example, the small-signal deviation about the operating point is denoted by \(\Delta \boldsymbol{u}_\mathrm{g} = \boldsymbol{u}_\mathrm{g} - \boldsymbol{u}_\mathrm{g0}\), where \(\boldsymbol{u}_\mathrm{g0}\) is the operating-point quantity. From (1)–(3), the linearized model for the estimation-error dynamics is obtained as

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    +(6)#\[\begin{split}\frac{\mathrm{d}\Delta \tilde{\boldsymbol{u}}_\mathrm{g}}{\mathrm{d}t} &= -\alpha_\mathrm{o}\Delta \tilde{\boldsymbol{u}}_\mathrm{g} + \mathrm{j}\boldsymbol{u}_\mathrm{g0} (\Delta \omega_\mathrm{g} - \Delta \hat{\omega}_\mathrm{g}) \\ +\frac{\mathrm{d}\Delta \hat{\omega}_\mathrm{g}}{\mathrm{d}t} &= k_\omega\mathrm{Im}\left\{ \frac{\Delta \tilde{\boldsymbol{u}}_\mathrm{g}}{\boldsymbol{u}_\mathrm{g0}} \right\}\end{split}\]
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    where \(\Delta \tilde{\boldsymbol{u}}_\mathrm{g} = \Delta\boldsymbol{u}_\mathrm{g} - \Delta \hat{\boldsymbol{u}}_\mathrm{g}\) is the estimation error.

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    First, assume that the grid frequency \(\omega_\mathrm{g}\) is constant and the frequency tracking is disabled. From (6), the closed-loop transfer function from the PCC voltage to its estimate becomes

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    +(7)#\[\frac{\Delta\hat{\boldsymbol{u}}_\mathrm{g}(s)}{\Delta\boldsymbol{u}_\mathrm{g}(s)} = \frac{\alpha_\mathrm{o}}{s + \alpha_\mathrm{o}}\]
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    It can be realized that both the angle and magnitude of the PCC voltage estimate converge with the bandwidth \(\alpha_\mathrm{o}\).

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    Next, the frequency-tracking dynamics are also considered. From (6), the closed-loop transfer function from the grid angular frequency to its estimate becomes

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    +(8)#\[\frac{\Delta\hat{\omega}_\mathrm{g}(s)}{\Delta\omega_\mathrm{g}(s)} += \frac{k_\omega}{s^2 + \alpha_\mathrm{o}s + k_\omega}\]
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    Choosing \(k_\omega = \alpha_\mathrm{pll}^2\) and \(\alpha_\mathrm{o} = 2\alpha_\mathrm{pll}\) yields the double pole at \(s = -\alpha_\mathrm{pll}\), where \(\alpha_\mathrm{pll}\) is the frequency-tracking bandwidth.

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    This PLL in estimated PCC coordinates is implemented in the class motulator.grid.control.PLL. The Grid-Following Control examples use the PLL to synchronize with the grid.

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    References

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    +By Aalto Electric Drives +

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    + + \ No newline at end of file diff --git a/docs/drive_examples/flux_vector/index.html b/docs/drive_examples/flux_vector/index.html index 3dcb6769..a1946595 100644 --- a/docs/drive_examples/flux_vector/index.html +++ b/docs/drive_examples/flux_vector/index.html @@ -229,7 +229,8 @@
  • Grid Converters
    • diff --git a/docs/drive_examples/flux_vector/plot_flux_vector_im_2kw.html b/docs/drive_examples/flux_vector/plot_flux_vector_im_2kw.html index df6ea489..92881f9c 100644 --- a/docs/drive_examples/flux_vector/plot_flux_vector_im_2kw.html +++ b/docs/drive_examples/flux_vector/plot_flux_vector_im_2kw.html @@ -229,7 +229,8 @@
  • Grid Converters
    • @@ -489,7 +490,7 @@

    2.2-kW induction motor

    plot(sim, base)
    -plot flux vector im 2kw

    Total running time of the script: (0 minutes 9.817 seconds)

    +plot flux vector im 2kw

    Total running time of the script: (0 minutes 10.085 seconds)